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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z(Jrich
182 Leonard D. Baumert California Institute of Technology Pasadena, CA / USA
Cyclic Difference Sets
Springer-Verlag Berlin. Heidelberg- New York 1971
I S B N 3-540-05368-9 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05368-9 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data hanks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag Berlin - Heidelberg 1971. Library of Congress Catalog Card Number 73-153466 Printed in Germany. Offsetdruck: Julius Beltz, Weinheim/Bergstr,
CYCLIC DIFFERENCE SETS
A fairly comprehensive survey of the general theory of cyclic difference sets is given below°
The aim of ~his survey is to provide a cohesive presentation of
the known facts as well as an introduction to some of the outstanding problems. The more general topics of block designs and difference sets in finite groups are introduced but only those aspects of these subjects which shed some light on problems arising for cyclic difference sets are developed. It is not expected that many will wish to read this survey sequentially from the beginning.
For this reason the chapters and to a lesser degree the sections
within them are largely independent of each other, having been written that way in order to encourage the reader to skip around and follow his own interests.
However
a certain familiarity with the contents of Chapter I is presupposed elsewhere. Beyond this, interconnections between the various sections and chapters are indicated when they seem relevant.
This structure~
coupled with the aim of making the
later material understandable to as many as possible, has led to the anomaly that, in some cases, quite elementary concepts are defined in the later chapters, whereas these
same
concepts,
and a great deal more, were presupposed in earlier sections.
In addition to the specific references inserted in the text, the books of Marshall Hall, Jr., "Combinatorial Theory", Blaisdell Publishing Company, 1967, of H. B. Mann, "Addition Theorems", Interscience Publishers, "Combinatorial Mathematics",
1965, and of Ho J. Ryser,
Carus Mathematical Monograph No. 14, 1963, may be used
as general references for a large part of this material° This survey was compiled in connection with research carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under Contract No.
NAS 7-100, sponsored by the National Aeronautics and Space Administration.
CONTENTS I.
II.
III.
IV.
V.
INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . .
A.
Difference
B.
Shifts~
C.
Block Designs,
D.
The Characteristic
E.
Multipliers
F.
The Hall-polynomial,
G.
Group Difference
EXISTENCE
Sets
i
. . . . . . . . . . . . . . . . . . . . . .
Equivalence
Complements
. . . . . . . . . . . . . . .
Incidence Matrices. Function
i
The Incidence
Equation . . . . .
and its Autocorrelation
Function
....
. . . . . . . . . . . . . . . . . . . . . . . .
QUESTIONS
w-multipliers
4 6 7
. . . . . . . . . . . . . .
Sets . . . . . . . . . . . . . . . . . . . .
8 9
. . . . . . . . . . . . . . . . . . . . . .
ii
A.
The Main Existence
B.
The Bruck-Ryser-Chowla
Co
Integral
D.
The Theorem of Hall and Ryser . . . . . . . . . . . . . . . . .
24
E.
Results of Mann,
. . . . . . . . . . .
26
. . . . . . . . . . . .
54
MULTIPLIERS
Problems
2
Solutions
. . . . . . . . . . . . . . . . .
Theorem
. . . . . . . . . . . . . . . .
to the Incidence
Rankin,
Equation
Turyn and Yamamoto
A N D CONSTRUCTIVE
ii
EXISTENCE
TESTS
12
. . . . . . . . . .
18
A.
Multiplier
Theorems . . . . . . . . . . . . . . . . . . . . .
54
Bo
Difference
Sets Fixed by a Multiplier
. . . . . . . . . . . . .
60
C.
Multipliers
. . . . . . . . . . . . .
62
D.
Polynomial
DIFFERENCE
and Diophantine Congruences
Planar Difference
B.
Hadamard Difference
C.
Barker Sequences.
Ao
Sets
. . . . . . . . . . . . . . . . .
.
N th Power Residue
.
77 77
Sets . . . . . . . . . . . . . . . . . . .
90
SETS
Hadamard Matrices . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
Singer Sets and Their Generalizations. .
65
. . . . . . . . . . . . . . . . . . .
Circulant
OF DIFFERENCE
and Welch . B.
. . . . . . . . . . . . . . . . . . .
SETS OF SPECIAL TYPE
Ao
FAMILIES
Equations
.
.
.
.
Difference
.
.
.
The Results
of Gordon,
99 Mills
. . . . . . . . . . .
Sets and Cyclotomy
96
. . . . . . . . .
99 119
V.
VI.
FAMILIES OF DIFFERENCE C.
More Cyelotomic
D.
Generalized
MI SCELLANY
SETS
(continued)
Difference
Sets
. . . . . . . . . . . . . . .
Cyclotomy and Difference
Sets
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
A.
Multiple
Inequivalent
B.
Searches
C.
Some Examples
D.
A Table of Difference
Difference
Sets
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sets
. . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . .
127 131 1 43 143 14h 146 148 159
I.
INTRODUCTION
The main purpose of this chapter is to provide the basic definitions and vocabulary of the study of difference sets so that subsequent chapters need not be interrupted at inopportune times b y the introduction of such material.
Thus none
of the concepts are pursued in detail - such development being deferred to the appropriate part of a later chapter.
A.
Difference Sets A
v, k, ~ - difference set
modulo
v,
D = {dl, o.., ~ }
such that for an~ residue
J # 0
d i - dj ~ ~
has exactly
h
solution pairs
(di,dj)
changeable below;
(mod v)
k
residues
the congruence
(mod v)
with
difference set, cyclic difference set and
is a collection of
di
(ioi)
and
dj
in
Do
[The terms
v, k, h - difference set are inter-
the later two being used when there is some reason to stress
either the contrast with general group difference sets or the particular parameters v, k, k
involved.]
As an Lv~ediate consequence of this definition one has that the relation
(1.2)
k(k- 1): ~(v- i)
necessarily holds among the parameters Given any positive integer modulo
v. (i) (ii)
(iii) (iv)
v
v, k, ~.
there are certain obvious difference sets
These are: the null set
D =
all singletons O = [O,1 ..... v -
D = [i},
0 < i < v - 1
l}
D = [0,i ..... i - I, i + i, .... v - i]
0 0,
sets
for any integers
t, s
equation
difference
sets.
(1o2) there may or several in-
In fact, Gordon, Mills and
difference
in V. A. below)°
it is
are two difference
there exist values of
inequivalent
set~
the
is also a difference
set (up to equivalence)
sets having these parameters.
that there are at least
Dj
s
Again, given any integer
are said to be equivalent
sets, a single difference
difference
Di
D.
taken modulo
D. = tD. + s l j
Welch (1962) have shown that given any
meters.
If
then for any integer
v, k, h
such
sets with these para-
Among the examples given in I. A. above note that
D1
and
D2
are related~
that is
D I + D 2 = {O,l,o..,v - I} o
For this reason D
D1
is a difference
and
set.
v*, k*~ v
and
For m o s t purposes
m e n t a r y difference than
v/2.
are said to be complementary difference
set w i t h parameters
set w i t h parameters one sees that both
D2
h*= n
D*
is a difference
Recalling that
n = k - h
it is sufficient to consider only one of a pair of comple-
sets.
This is frequently done by insisting that
of the four - v, k, h, n. Using
k = v/2
k
be less
does not occur)°
the parameters
v, n
are the most fundamental
For this reason it is sometimes u s e f u l to express (1.2) and
n = k - h
k " k* = k(v - k) = n(v - i)
h + h* = v - 2n
its complement
If
are invariant under complementation of the difference
As w i l l become abundantly clear,
Since
h
v, v - k, v - 2k + h.
(Equation (1.2) shows that
in terms of them.
v, k,
sets.
and
k > 1
and
k, h
it follows that
h. h* = h(v - 2n - h) = n(n - i) .
for non-trivial difference
sets~
it follows
that
_>
n(n-
> v-
2n- i
and so
n
The two extreme cases
2
+n
v : 4n - i
+ i >v>4n-
and
v = n2 + n + i
sets of the Hadamard type and to the difference projective planes
io
(so-called planar difference
cussed more fully in Chapter IV - Difference
correspond to difference
sets associated with certain finite sets).
Both these types are dis-
Sets of Special Type.
C.
Block Designs, Incidence Matrices, An arrangement of (i) (ii)
(iii)
v
The Incidence Equation
objects into
b
sets (called blocks) such that:
each block contains exactly
k
different objects
each object occurs in exactly
r
different blocks
any pair of objects occurs in exactly
is called a balanced incomplete block design.
~
different blocks
Evidently the relations
b k = vr
(1.3)
r(k - i) = h(v - i)
are satisfied b y the parameters
b, v, r, k, h.
pairs involving any one fixed object.) number of objects equation
v
(1.4)
(The last follows by counting the
If the number of blocks
the design is said to be s~mmetric.
(1.3) shows that
k = r.
b
equals the
In a symmetric design
Thus for symmetric designs equation
(1o4) takes
the same form as equation (1.2). Every difference set gives rise to a symmetric block design whose objects are 0,1,...,v - 1
and whose blocks are
D,D + 1,...,D + (v - 1).
Note that any cyclic
relabeling of the objects of such a design permutes the blocks cyclically in a cycle of length
v.
Such a block design is called cyclic.
Thus every difference set
corresponds to a cyclic sy~netrie balanced incomplete block design. the difference set
D1
of
For example,
I.A. above corresponds to the block design:
BI = DI
- [i, 2, 4]
B 2 = D I + i =- [2, 3, 5] B3 = D I + 2 - [3, 4, 6]
B4 = D I + 3
- I0, 4, 5]
B 5 = D I + 4 - [i, 5, 6] B 6 = D I + 5 -= [0, 2, 6] B 7 = D I + 6 -= [0, l, 3]
(rood 7)
Associated with any balanced incomplete block design is a zeros and ones called the incidence matrix of the design. putting wise.
aij = 1
b × v
For the example above [recalling that the ith block is j - l]
A =
of
A
It is constructed by
if the jth object appears in the ith block and
that the jth object is
matrix
aij = 0
D + (i - l)
otherand
the incidence matrix is
0 0 0 i 0 i l
1 0 0 0 1 0 l
1 1 0 0 0 1 0
0 1 1 0 0 0 1
1 0 1 1 0 0 0
0 1 0 1 1 0 0
0 0 1 0 1 1 0
From the block design definition above it is clear that the associated incidence matrix satisfies the so-called incidence equation
ATA:
where
I
is the identity of order
same order.
(r-
v
and
(1.5)
:):+
J
is the matrix of all ones of that
From this it follows that the determinant of
ATA
is given by
:ATAI = [r + (V - l)~] (r - :):-i
(l.6)
If the block design is symmetric, this becomes (using equation (1.4))
IATAI = k2(k - ~)v-i
from which it follows that the incidence matrix balanced incomplete block design is non-singular.
A
(1.7)
of a non-trivial symmetric Using this fact it can be shown
(Ryser, 1950) that the incidence matrix of a syn~netric block design is normal. is, that
That
(1o8)
ATA = AA T = (k - ~,)I + ~7
for symmetric designs. Since
IATAI = IAI 2
one can conclude from equation (1.8) or (1.7) that a
symmetric block design (or a difference set) can only exist for even n = k - ~
is a square.
v
if
The study of equation (1.8) has led to other significant
existence criterions for symmetric block designs and hence for difference sets. Since Chapter II is devoted to difference set existence questions these results are deferred until then.
D.
The Characteristic Function and its Autoeorrelation Function Corresponding to the difference set
(i = 0 , . . . , v wise.
l)
given by
a. = 1 1
if
D i
is the binary sequence is a member of
D
and
{a i] a. = 0 1
other-
This is called the characteristic function or incidence vector of the
difference set
D.
[Of course it appeared in I. C. as the first row of the
incidence matrix associated with the difference set. ]
Considered as a binary
sequence, it is quite natural to inquire about its autocorrelation function v-1 Ra(J) =
~
a i ai+ j
(i + j
taken modulo
v)
i=O
and normalized autocorrelation function
Pa(j)~ !v Ra(J)' Since
{a i]
is the characteristic function of a difference set, the autocorrelation
ftunction is particularly simple, i.eo
Ra(J) = I k
if h
j -= 0
otherwise
modulo
v
Frequently the binary sequence
fb " i }v-1 0
where
b i = 2a i - 1
is considered
instead of the characteristic function (note the transformation merely replaces the zeros of
[a i}
with minus ones).
This sequence
{b i]
has autocorrelation
function
J
v
if
j ~ 0
\
V - 4(k - ~)
modulo
v
otherwise
Autocorrelation functions like these are said to be two-level and binary sequences which possess them have found extensive application in digital communications. [See Golomb et al, 1964 for some of these].
Of course, it follows immediately from
the definitions that the only binary sequences which have two-level autocorrelation functions are those associated with cyclic differences sets.
E.
Multipliers If
t
is prime to
difference set
D,
then
v
and if t
tD
is some shift
is called a multiplier of
D + s D.
of the original In terms of the
associated block design of the difference set (see I.Co above) the mapping (mod v)
is an automorphism.
That is~ if
A
is the incidence matrix of the associ-
ated block design, there exists permutation matrices order
v
x~tx
P, Q
determined by
t
of
such that
PAQ = A .
All known multipliers
(1°9)
v, k, h - difference sets have non-trivial multipliers
t ~ 1
modulo v).
(i.e.,
The question as to whether this must be so is open.
The collection of multipliers of a given difference set forms a group called the multiplier group of that difference set.
One of the most useful results in the
theory of difference sets is the so-called "multiplier" theorem.
This theorem
guarantees the existence of multipliers under certain circumstances.
Theorem i.io is prime to
v
(Hall and Ryser~ 1951 ).
and if
with these parameters
p > h,
then
p
If
p
is a prime dividing
n,
is a multiplier of all difference sets
v, k, h.
Chapter III discusses this result and its generalizations.
For the present,
merely note that for all known non-trivial difference sets the condition superfluous.
p > h
is
Again since (another open question) all known non-trivial cyclic
difference sets have
(n,v) = 1
difference sets every divisor
F.
if
one has that for all known non-trivial cyclic t
of
n
is a multiplier.
The Hall-polvvnomial, w-multipliers Instead of the difference set itself, it is often convenient to deal with the
polynomial
e(~):x
dl
+...
%
+x
.
(l.lO)
This pol~nqomial has been called the Hall-pol~nomial of the difference set, the generating polynomial of the difference set or the difference set pol~omial. In terms of this polynomial the difference set property is
e(x)e(x-1)
k ~. =
x
d. -d. I j
=
n
+
h(1 + x
xV_l ) +
o"
+
(mod x v - i) o
(l.ll)
i,j
If
~v ~ 1
is any vth root of unity this congruence yields
(1,z2)
e({v) 8({v I) = n.
This shows that a non-trivial difference set is intimately connected with the factorization of If
t
n
in the field of vth roots of unity.
is a multiplier of the difference set
D
then
e(x t) ~ xSe(x)
More generally, prime to
w,
if
w
divides
v,
(mod x v- l) .
define a w-multiplier to be any integer
for which there exists an integer
e(x t) ~ xSe(x)
Clearly
s
s
(mod
wI
w-multiplier for every divisor
w
of of
w.
t,
satisfying
xw - l ) .
may be assumed to be non-negative.
wl-multiplier for all divisors
(1.13)
(1.14)
Further, a w-multiplier is a
Thus, an ordinarymultiplier
is a
v.
On occasion it is possible to demonstrate that a hypothetical difference set must have a w-multiplier even when it is not possible to show the existence of a multiplier.
This often leads to non-existence proofs which can not be deduced
from the strict multiplier theory.
G.
Group Difference Sets A difference set in a group
distinct elements of
G
G
of order
v
is a set
{gl~..°~gk]
of
such that the equation
-1 gigj = g
has exactly
h
solutions for every
g
in
G, g ~ 1.
Group difference sets are called non-Abelian, Abelian~ or cyclic according to whether the group is non-Abelian, Abelian or cyclic.
The difference sets considered
above (I.A., etc.) correspond to group difference sets for cyclic groups (i.e., they are cyclic difference sets under this terminology). sets constitute a generalization consideration.
Thus group difference
(due to Bruck, 1955) of those previously under
Every group difference set gives rise to a symmetric block design
in much the same manner as demonstrated for cyclic difference sets in I.C. above. But not every symmetric block design corresponds to a difference set in some group G.
[For example, the
v, k, h = 31, 10, 3
design (listed in Hall, 1967, p. 293)
l0
could only correspond to a cyclic difference set since the only group of order 31 is cyclic°
But this design is obviously not cyclic. ]
occupy a t r ~ y
Thus group difference sets
middle ground between symmetric block designs and the cyclic dif-
ference sets of concern here. The main reason for introducing general group difference sets into this discussion at all is that some of the major outstanding problems are only of concern for cyclic difference sets.
Thus, in subsequent chapters a few facts about
general group difference sets are mentioned in connection with these problems.
The
purpose being to point out that the difficulties arise only because of the cyclic nature of things and thus cannot be resolved solely by techniques which apply more generally. An example of a group difference set, which is not cyclic, is the set D = {a, b, e, d, ab, ed] where
a
2
= b
2
= c
2
=
d2
in the Abelian group of order i6 generated by = i.
This set has parameters
a, b, c, d,
v, k, )~ = 16, 6, 2.
II.
EXISTENCE QUESTIONS
The main questions regarding difference sets are: do you construct them?
When do they exist?
How many (inequivalent ones) are there?
How
Even though there
is considerable overlap between the areas defined by these questions, there seems to be some value in treating them separately.
Thus, this chapter is primarily
concerned with conditions necessary for the existence of difference sets. most part only number theoretic results involving the parameters their divisors are considered.
For the
v, k, ~, n
and
Some existence tests of a more constructive nature
are presented in Chapter I I I - Multipliers and Constructive Existence Tests.
A.
The Main Existence Problems As the title of this chapter implies, the existence question for difference
sets is unsolved.
That is, given parameters
v, k, ~
it is (in general) impossible
to decide (short of an exhaustive search) whether or not a difference set with these parameters exists.
Nevertheless,
significant progress has been made.
Perhaps the most important test is the obvious relationship
k(k - i) = ~(v _ i) or
k2
= ~v + n .
(2.1)
A sub-problem of this general existence question is the curious fact that no difference sets are known which have
(v,n) > 1
though no proof of this has been given.
[or equivalently
Here one must be careful to distinguish
between cyclic difference sets and general group difference sets. exist group difference sets with parameters with
(v,n) > l;
v, k, k, n = 16, 6, 2, 4.
(v,n) > l,
(k,v) > 1],
For~ there do
the example given in I.G. above has
Thus, if there are no cyclic difference sets
the proof must be intrinsically cyclic.
12
Another outstanding existence problem arises when one notes that there exists an infinite number of difference sets with pJ + l, pJ + l, 1 struction).
for all primes p
Z = 1.
Specifically
(see Section V.A. for details of their con-
It has been conjectured that for every
finite number of difference sets.
v, k# k = p2j +
Z ~ 2,
there exists only a
This conjecture is wide open; it has not been
either proved or refuted for any single value of
Z.
Of course the same conjecture
can be made for symmetric block designs and again the problem is open.
B.
The Bruck-Ryser-Chowla Theorem As pointed out in Section I.C. above, the Incidence Equation~
i.e.,
ATA = nI + ~J
holds for all symmetric block designs. of the design, order
v).
J
is the
v × v
(Here
A
(2.2)
is the
matrix of all ones and
v x v I
incidence matrix
is the identity of
Associate a linear form with each row of the incidence matrix
A~
according to the rule
v Li(x ) =
~
aij x. J
j=m
where
x = (Xl, .... Xv)
takes the form
is a vector of indeterminates
x.. J
Then equation (2.2)
13
L~(X) + ... + L ; ( X ) =
n(~
+ -.. + 4 )
(2.3)
+ h(Xl + "'' + Xv)2
The study of these equations ((2.2) and (2.3)) has produced a number of existence criterions for symmetric block designs as well as for certain more specialized configurations. Let
B = nI + ~J
and write
ATA
as
ATIA.
Then (using the language of
quadratic forms) equation (2.2) shows that if a block design exists, then the identity matrix I represents fortunately, when
A
B
with a
0, 1
transformation matrix
the theory of such matrix representations
Un-
is not fully developed even
is allowed to have arbitrary integer coefficients.
permited rational coefficients,
A.
However, if
A
is
the Hasse-Minkowski theory of rational equivalence
of quadratic forms [see Jones (1950) for an exposition of this theory] provides necessary and sufficient conditions for the existence of such a transformation
A.
Specifically
Theorem 2.1 (Bruck-Ryser-Chowla) ATA = nI + ~J,
when
k(k - l) = h(v - 1),
(i)
for
v
even,
(ii)
for
v
odd, the equation
integers
A v x v
n
x, y, z
rational matrix
A
satisfying
exists if and only if
is a square z
2
= nx
2 +(-i )(v-l)12y2
has a solution in
not all zero.
Thus Theorem 2.1 provides necessary conditions for the existence of symmetric block designs and hence for difference sets.
In f a c %
there is no parameter set
v, k, ~
satisfying Theorem 2.1 for which it is known that no symmetric block design
exists.
That is, conceivably the conditions of Theorem 2.1 are sufficient not only
for the existence of a rational matrix
A
but also a
O, 1
matrix.
If one
restricts attention to cyclic difference sets however, this is no longer the case. For [as is shown later, section II.E] there is no cyclic difference set with parameters
v, k, k = 16, 6, 2
even though these parameters do satisfy Theorem 2.1,
14
as they clearly must since an example of such a block design can be derived from the non-cyclic difference set given in section I.G. above. It should be pointed out that Legendre
[see Nagell (1951) Theorem 113] provided
a simple effective test for the solvability of diophantine equations of the type appearing in (ii) above (see Note 2 below for a statement of this test).
Thus
criterion (ii) is an effective criterion even when a solution of the diophantine equation is not obvious. consider
As an example of a parameter set excluded by this theorem
v, k, k = 43, 7, i;
this leads to the diophantine equation
z 2 = 6x 2 _ y 2
Necessity of the conditions of Theorem 2.1 can be proved without recourse to the Hasse-Minkowski theory (see below). h = i n
Even sufficiency is available for the case
[Hall (1967), p.lll] and also whenever
n
is a square.
In particular, when
is a square the rational matrix
A= Jni+
gk n j _ _ v
satisfies equation (2.2).
Proof of Theorem 2.1
[necessity only, Chowla and Ryser (1950)].
(2.2) it follows that
(det A) 2 = k2(k - h)v-l;
is a square.
v,
For odd
thus when
v
is even,
From equation n = k - h
the number-theoretic result that every positive integer
is representable as a sum of four integral squares
[see, for example, Nagell (1951)]
and Euler's identity
(b
where
2 2 2 2 2 2 ~ 2 2 2 2 + b 2 + b 3 + b4)(x I + x 2 + x 3 + x ) = Yl + Y2 + Y3 + Y4
(2.4)
15
Yl = blXl - b2x2 - b3x3 - b4x4 Y2 = b2xi + blX2 - b$x3 + b3x4
(2.5) Y3 = b3Xl + b4x2 + blX3 - b2x4 Y4 = b4Xl - b3x2 + b2x 3 + blX4
are required.
system of equations solved for the efficients, When Xv = YV
n = b 2I + b 22 + b 32 + b 24
With
(2.5) is
x.'sl
n 2.
Thus Cramer's
as linear combinations
the denominators v ~ i (mod 4),
integers) the determinant
(b i
of which are
the relation
rule shows that the system may be
of the
Yi'S
with rational co-
2
n .
2 2 2 n = bI + b2 + b3 + b
together with
and
2 2 2 2 2 + 2 2 n(x2 + Xi+l + xi+2 + xi+3) = Yi + Yi+l Yi+2 + Yi+3
for
i = 1,5,...,v- 4
the independent
can be used to transform equation
indeterminants
yl,...,y v
Ll(Y),...,Lv(Y )
yl,...,y v. YI'
and
given by
w = x I + x 2 + .-o + x v
Since (2.7) is an identity in the
Yi'S
(2"7)
are rational linear forms in it is valid for all values of
in particular for the value
cev2
+ ..o + C J v
for i-
cI
cI ~ 1
(2.8)
Yl = e2Y 2 + ..° + C J v
for -i-
where
(2.6)
(2.3) into an identity in
2 2 2 7,~w2 L (y) + ... + L (y) = Yl + "'" + Yv-I + nYv +
where
of the
cI
cI = 1
16
n
LI(Y) =
Z
cjyj .
J=l
For this value of
Yl
however,
identity in the variables
L~(y) = y~;
Y2, o.o,yv.
thus equation (2.7) reduces to an
Proceeding in this manner with
y2,...,yv. 1
in turn, yields the identity
L2v(y) = ny2v + %~w2
where
Lv(Y )
integer
x,
and
w
are rational multiples of
Yv"
Now let
Yv
be a non-zero
which is a multiple of the denominators appearing in
then in integers
Lv
and
w,
x, y, z (x ~ 0) the equation
z2 = n x 2 + by2
(2.9)
has a solution. When Xv+ I
v ~ 3 (mod 4),
add
is a new indeterminate.
nx
2 v+l
to both sides of equation (2.3) where
Proceeding as before yields the identity
! 2 2 = 2 2 L (y) + ".. + Lv(Y ) + nXv+ I Yl + "°" + Yv+l + hw2
where
Ll(Y),o..,Lv(Y), Xv+ I
Again choosing
yl,...,y v
and
w
are rational linear forms in
yl,...,yv+ 1.
judiciously implies the identity
2 = 2 nXv+l Yv+l + hw2
where
Xv+ I
and
w
are rational multiples of
non-zero integer
z
which is a multiple of the denominators of
yields a solution in integers
Yv+l"
x, y, z (z ~ O)
2
2
Taking
of the equation
2
Yv+l Xv+ 1
to be a and
w
17
Combining this equation with that of (2.9) completes the proof of the necessity of the conditions of Theorem 2.1. Condition (i) of Theorem 2.1 was derived independently by Sch[tzenberger and by Chowla and Ryser (1950).
Condition (ii) was first established,
only, by Bruck and Ryser (1949) and then generalized to arbitrary
~
for
(1949)
h = 1
independently
by Chowla and Ryser (1950) and Shrikhande (1950). As pointed out in section I.C. the incidence matrix
A
of a symmetric block
design not only satisfies equation (2.2) above but also must be normal (Ryser, 1950)o
That is, it must satisfy
(2.io)
AA T : ATA = (k - h)l + ZJ.
Thus, when Theorem 2.1 was first established,
there was some reason to hope that
adding the normality condition would further restrict the possible parameter sets v, k, h.
This was shown not to be true by Albert (1953) for
and Ryser (1954) for general
h.
k = i
That is, the conditions of Theorem 2.1 are
sufficient to guarantee the existence of a normal rational matrix equation (2.2).
Moreover,
and by Hall
this solution
A
A
satisfying
also satisfies the condition
AJ = kJ
which is trivially necessary for block designs° Hall and Ryser also showed that any set of initial
0, i
rows, consistent
with equation (2.10) above, can be completed to a normal rational matrix satisfying that equation. more general result.]
[See Hall (1967) p. 275 for the proof of a somewhat
Clearly, specifying an initial set of
to equation (2.10) yields the same result. this to the case where
r
rows and
s
of equation (2.10) and those imposed by for symmetric block designs. rational matrix Note i.
A
A
%1
columns subject
Eo C. Johnsen (1965) has generalized
columns are given, subject to the provisions AJ = JA = kJ,
which are clearly necessary
He showed that even here there always exists a normal
satisfying equation (2.10) as well as
AJ = JA = kJ.
The definition of the incidence matrix of a block design varies with
different authors and even with different works of the same author. the matrix designated by
AT
Some say that
above is the incidence matrix instead of
A.
Since
18
symmetric block designs yield normal incidence matrices for the designs themselves. proofs.
But it is sometimes
Note 2.
a factor in the mechanics
If
has a solution
a, b, c
are squarefree
in integers,
quadratic
residues
quadratic
residue of
ax
of
integers which are relatively prime in
2
+ cz
2
m
= 0,
without
be divided out. i,
2
divide that
ez ~ g
there
but since
m a y be divided out. manner yielding
Co
Integral
above.
significant
z.
represents
x
g
is said to be a
such that
Given a ~
are
x
2_
= g
diophantine
equation whose
modulo
equation coefficients
Then,
any factor common to with greatest
a, b, c
con~non divisor
if the equation has a solution
are squarefree
and
(a,c) > i
or
(a~ b, c) = 1 ax
2
+ by
(b,c) > i
2
can
g
must
this implies
22 + cg z I = O.
can be handled
Thus
g
in the same
an equation of the desired type.
Solutions
to the Incidence
are discussed
section II.D.
-ac, -ab
(g,m) = l,
are squarefree
Thus the equation becomes Clearly
-bc,
diophantine
Similarly
a~ b, c
a, b
ax 2 + by 2 + cz 2 = 0
Clearly any square factor m a y be dropped from
change.
Since no further existence figurations
[If
other~ise. ]
is an associated
(a~b) = g > 1.
divides
if and only if
respectively.
nonresidue
Thus assume
and assume
equation
if there exists an integer
satisfy the restrictions a, b, c
not all zero,
a, b, c
and called a quadratic + by
of the
a reference.
pairs and not all of the same sign~ the diophantine
2
in no errors
Thus one should be careful to check the incidence m a t r i x definition being
used when consulting
m
this results
Equation
tests for difference
in this section,
The material
sets or more general
the reader m a y wish to skip to
cited here not only is of independent
a survey of the present
con-
status of the integral
interest but also
solution problem;
one
would hope that a complete
solution of this problem would provide new existence
criteria for block designs
and difference
As noted previously and symmetric matrices
sets.
there is a natural correspondence
which links each form
between
quadratic
forms
19
n
ci$.x.x. I ~
f(x) =
c.lj = c.. Sm
(2.11)
i,j=l
with the matrix of its coefficients
C = (cij).
are restricted to the real numbers.
Given two n-variable forms
their matrices
C, D, f(x)
transformation taking
In this discussion the coefficients
is said to represent
f(x)
into
g(y).
g(y)
f(x), g(y)
and
if there exists a linear
That is, if there exists a substitution
n xi : ~
sijY j
i = l,.o.,n
j=l
transforming
f(x)
into
g(y).
If
S = (sij),
this means that
s~cs : D
in which case
C
efficients then
is said to represent f(x)
D.
(2.12)
If
is said to represent
S
g(y)
is restricted to rational corationally
(C
represents
rationally) with analogous statements for other coefficient restrictions. f(x)
represents
integer
m~
g(y)
D
Further,
rationally without essential denominator if, for every
there is a matrix
S
of rank
rational elements with denominators prime to
n
such that m.
sTcs = D~
If two forms
where
f(x), g(y)
S
has
repre-
sent each other rationally without essential denominator they are called semiequivalent and said to belong to the same genus. If
f(x)
determinant
represents ± I,
g(y)
integrally and the transformation matrix
has
then clearly
g(y)
represents
formation matrix of determinant
± i.
Two such forms are called equivalent and
said to belong to the same class.
f(x)
S
integrally with trans-
Thus equivalent forms are clearly semi-equivalent;
in fact a genus will in general contain several classes of forms. Turning back to block designs again, consider the incidence equation
20
ATA= (k- ~)Z + ~ = B .
This says no more than that the form v x v
2 2 f(x) = x I + ... + x v
with a
0, 1
(associated with the
g(y) = (k - h) (y~ + .-- + y~) +
identity matrix I) represents the form
h(Yl + "'" + Yv )2
(2.13)
transformation matrix
A.
The Bruck-Ryser-Chowla Theorem (Theorem 2.1 above) establishes necessary and sufficient conditions for the existence of a rational matrix equation (2.13).
A
satisfying
Unfortunately the theory of integral representations of quadratic
forms is not yet complete, even though a great deal of work has been done on such problems
[see Jones (1950), Watson (1960) and O'Meara (1963) for this work.]
How-
ever the main concern here is equation (2.13), not all possible integral representation problems, thus a complete theory is not required for these purposes. Goldhaber (1960) studied the integral representation problem posed by equation (2.13) for a restricted set of parameters
Theorem 2.2. (k,n) = l,
If
v, k, h
v, k, h.
He proved
satisfy the Bruck-Ryser-Chowla theorem and
then there exists a form in the genus of
I
which represents
B
integrally. However,
for
v ~ 8,
the genus of the identity matrix contains more than one
equivalence class of forms (Magnus, 1937), thus no immediate conclusion can be drawn regarding integral representation from this result. A
0, 1
matrix
A
satisfying equation (2.13) with
k(k - l) = h(v - l)
is
the incidence matrix of a symmetric block design~ although this is not immediately apparent. (i.e.,
For, with the incidence matrix definition of section I.C. above
a.. = 1 ij
if and only if block
i
contains object
yields the fact that every object occurs in exactly objects occurs in exactly contains precisely contain
k
b0~...,bv_ 1
h
distinct blocks.
k
equation (2.13)
blocks and that any pair of
So, the fact that every block
objects is yet to be established. objects respectively.
j),
Let blocks
B0,...,Bv_ 1
Then, from the conclusions already
drawn, it follows that
b 0 + b I + ... + by_ I = kv
21
b0(b 0 - I) + bl(b I - i) + "'" + bv_l(bv. 1 - i) = hv(v - i) = vk(k - i) .
So v-1
v-i
i=O
i=O
[bi(b i - i) - k ( k -
1)]
from which it follows that
v-i
v-i [bi(b i - i) - k(k - i) + b.l + 2k2 - k - 2kb.l ] = 0 .
i=O
But
i=O
Z(b i - k) = O = Z(b i - k) 2
block contains precisely Note that if
A
one of these can be a
Theorem 2.3.
k
implies that
objects,
b.I = k
satisfies equation (2.13) then so does 0, i
matrix.
then
A
or
i.
That is, every
as was to be proved. -A.
Of course, only
Ryser (1952) has shown:
If a normal integral matrix
k(k - I) = h(v - i),
for all
-A
is a
A
0, i
satisfies equation (2.13), for matrix and hence is the incidence
matrix of a syn~netric block design. As pointed out in section I.C. above, the incidence matrix of a symmetric block design is normal~ thus it satisfies: (i) (ii)
ATA = (k - ~)I + ~ AA T = (k - h)I + ~J
(iii)
AJ
: kJ
(iv)
JA
: kJ
Ryser has shown that a nonsingu!ar also one of (iii),
matrix,
satisfying one of (i), (ii) and
(iv) satisfies all four of them and also the relation
k(k - i) = h(v - !). needed;
v x v
[In Chapter V, the fact that (ii), (iii) imply (i), (iv) is
so a proof of this much of Ryser's result is included here.
Since
22
IBI = (k - h)v-l(vh - k + k), v h - h + k ~ O.
the nonsingularity of
A
means that
k - h ~ 0,
By assumption
AA T = (k - h)I + k/,
Multiplying this last by
A -1
shows that
AJ
k ~0
j2 = vJ.
kJ
and that
transposing both sides of this same equation yields Further, note that
=
,
A-Ij = k-Ij.
JA T = kJ,
since
Whereas j = jT
So
A T = A-!(AA T) = (k - ~)A -I + )~-ij = (k - h)A -1 + Ak-ij kJ = JA T = (k - ~)JA -I + hk-Ij 2 = (k - ~)JA -I + ~k-lvj.
Thus~
=
which serves to define
k-k
m.
Thus
k -
J = mJA
J =mJ
and
vJ = j2 = (mJA)J = mJ(AJ) = mJ(kJ) = mkvJ
which provides that
mk = i
JA -I = k-ij.
established.
or
Thus
m = k -i. J = k'ijA
As shown above
From or
JA -I = mJ
JA = kJ;
and
m = k "I
it follows
that is (iv) has been
A T = (k - ~)A -I + }~-ij;
multiplying through b y
yields
ATA
by (iv). promised.]
But this is (i).
=
(k - ~)I
+ k~-ljA
=
(k - ~)I
+
So (ii), (iii) have been shown to imply (i), (iv) as
Using a different part of Ryser's result, note that an integral
solution of equation (2.13) which also satisfies one of
AJ = kJ~ JA = kJ
is
A
23
normal and by Theorem 2.3 must be the incidence matrix of a symmetric block design. Let
A
multiplied
be a matrix through by
satisfying
-1,
Thus, given such a matrix
XX T = B,
the resultant A,
then if any column of
matrix
C
A
still satisfies
there is no loss of generality
is
XX T = B.
in assuming
that
its column sums are nonnegative.
Theorem 2.4.
(Ryser,
column sums,
satisfying
free and if
k - ~
1952)
If
AA T = B
A
with
is odd, then
A
is an integral matrix, k(k-
l) = Z ( v -
1),
with nonnegative
if
(k,Z)
is square-
is the incidence matrix of a symmetric block
design. The condition n = k - h
is even and
Theorem 2.5. v x v
k - h
Let
odd is necessary,
~ = !.
v = n
2
matrix w i t h nonnegative
of two types: even,
(i)
A
consist of
have sum zero.
In particular
n + 1
A
column sums satisfying
has a single
entries
There exists
an
+l
of an Hadamard matrix and also for
n = 10.
The orders of Hadamard matrices are limited to
IV. B. for more information
on this
In addition to the solutions
n
block designs
n = 2a
v = n
2
zeros~
n = 1,2 t
and
4t
be an integral A
is of one
entries
n
columms
which is the order
of order
for positive
n
with
integers
t.
subject].
of type
(ii) solutions
+ n + l, k = n + l, h = 1
is
+l, n + 1
and the remaining
(ii) given above, for the case
E. C. Johnson
(1966A)
n -= 2 (mod 4).
there are known to be both types of solutions;
that type
n
[see the note at the end of section
always exist as well as Hadamard matrices
b e e n conjectured
A then
n2 + n
(ii) for every
has found infinitely m a n y type (ii) solutions when
2
(1, -1 square matrices
It is not known whether they exist for all
Further,
and let
AA T = B,
zero and
and
of type
+ n n/2)
[see Hall (1967), p. 286].
+ n + l, k = n + l, ~ = 1
A
determinants
exist when
is the incidence matrix of a block design or (ii)
one of the columns of
columns
for many counter examples
of order
always exist for even
satisfy the Bruck-Ryser-Chowla
for such
n = 2 a. n
It has
whenever
condition.
D.
The Theorem of Hall and Ryser d1 If a cyclic difference
set exists,
then its Hall-polynomial
e(~)
:
x
+
-.-
% + x
satisfies the congruence
e(x)@(x -I) m n + ~(i + X + .o" + X v-l)
Thus, for any divisor
w
of
v,
(2.14)
(mo~ x w - i)
(2.i5)
(mo~ x w - i)
(2.16)
one has
w-1 @(x) ---b 0 + blX + .-- + bw_IX
e(x)e(x -I) ~ n +
(mod x v - l)
(1 + X + "'" + X w-l)
~v W
where
bi
is the number of
[equate coefficients
dj
in
in congruence
D
satisfying
w-i
~
b.2
=
n
+
1
~v
~
~
w
'
i=O
j = l,o..,w - i.
This implies
2.16]
w-1
for
d. -= i mod w. J
bibi_j
-
hv w
i=O
(Here the subscript
i-j
is taken modulo
w.)
So if
S
is the cyclic matrix defined by
S
b0
b I ... bw_ I
bI
b 2 ... b 0
=
bw. I b 0 ... bw_ 2
it follows that
ssT=
D
(2.17)
25
where
D
has
n + (hv/w)
on the main diagonal and
kv/w
in all other positions.
Applying the m e t h o d of Chowla and Ryser (see the proof of Theorem 2.1 above), it follows that in order for there to exist a rational matrix
S
of odd order
w
satisfying equation (2o17) it is necessary that the equation
Z2 = ~
have a solution in integers in this solution = z + kx
and
y ~ 0.
2
+
x, y, z,
(_l)(W-l)/2(~v/w) y2
not all zero.
If
(2.1s)
n
Thus, for this case, the integers
~ = y(hv/w)
is not a square, then ~ = zk + nx,
are not all zero and they satisfy the diophantine
equation
[2 = ~ 2 + ( - n ( w - 1 ) / ~ 2 .
Of course this equation always has a non-trivial solution when
(2.19)
n
is a square.
Thus
Theorem 2.6 for odd
v,
integers
(Hall and Ryser, 1951)
then, for every divisor
x, y, z,
w
If a non-trivial difference set exists of
v, equation (2.19) has a solution in
not all zero.
From the derivation above it is obvious that equation (2.19) has a non-trivial solution for w satisfied. only when
= v
whenever condition (ii) of the Bruck-Ryser-Chowla
Thus Theorem 2°6 can be stronger than the Bruck-Ryser-Chowla Theorem v
is odd and composite.
v, k, h = 39, 19, 9and
z 2 = 10x 2 - 39Y 2
respectively. z
2
Theorem is
Here with
w = v = 39,
and they have solutions
However, for
= 10x 2 + 13y 2
w = 13,
the two equations are x, y, z = l, l, 1
these parameters does exist however. 40;
z 2 = 10x 2 - 9 y 2 and
2, l, 1
the equation of Hall and Ryser is
and it has no non-trivial integral solutions.
difference set exists with parameters
matrix of order
An example is furnished by the parameter set
39, 19, 9.
Thus no cyclic
[A symmetric block design with
It is easily derived from any Hadamard
Hall (1967, Chapter 14) discusses construction methods for
26
Hadamard matrices.] Hughes (1957) used the work of Hall and Ryser to prove a more general theorem about the structure of certain symmetric block designs.
However, for difference
sets the result of Hughes says no more than Theorem 2.6 above.
Note°
Again only the conditions relating to rational solution of equation
(2.10) were developed, whereas it was known that b. < v/w 1
for
i = 0,1, o.o~W - 1.
S
was integral, cyclic and that
In particular perhaps some improvement in the
status of the integral representation problem for quadratic forms (see section II.C) would provide stricter existence tests.
E.
Results of Mann, Rankin z Turyn and Yamamoto By far the most stringent difference set existence tests yet discovered arise
from algebraic number theoretic consideration of the equation
e(~v) e(~v l) =
which must hold for a ~
v th root of unity
(2,20)
n
~v # i
if a difference set is to exist.
Most of the results cited below are attributed to Mann (1964)~ Turyn (1965) and Yamamoto
(1963) even though the work in Mann (1952 , 1965, 1967), Rankin (1964) and
Turyn (1960, 1961) is directly related.
In fact many of the results were discovered
independently by more than one of these authors°
A few of the theorems require the
knowledge of a multiplier or w-multlplier for the hypothetical difference set under question (see sections I.Eo and I.F. for these definitions).
Chapter III discusses
conditions on the parameters under which certain multipliers are known to exist° The results of this section are all stated in terms of the parameters v, k, h, n
and their divisors;
thus no algebraic number theory is required to
either understand the nature of the results or to apply them.
However,
knowledge of algebraic number theory is required for the proofs.
some
In order to
minimize this requirement some basic facts about cyclotomic fields (i.eo~ algebraic number fields generated by the roots of unity) are presented at the end of this
27
section (Theorems 2.18, 2.19 and 2.20). used.
Unless otherwise specified,
of unity and
K(~d)
~d
They will be referred to when they are will be an arbitrary primitive d th root
will denote the cyclotomic field it generates over the rational
field. If then b
a
a, b, c C
are integers
(c ~ O)
is said to strictly divide
and
b.
ac
divides
b
while
a c+l
does n o t ,
The same terminology is used if
a
and
are ideals°
Theorem 2.7 trivial
n
for which
modulo
Proof.
Let
strictly divide
w,
P
n
be a divisor of
~(~w).
v
and assume a non-
t > i.
Let
If there exists an integer
p
f > 0
is strictly divisible by an even power of
be a prime ideal divisor of
p
in
K(~w)
and let
be a prime such that p.
pi
Then by (2.47) of Theorem 2.19 below and equations pi
(1.14),
also strictly divides
Thus, by equation (2.20), n
w > i
(p,w) = i0
then
(2.20) it follows that
that
Let
v, k, h-difference set exists with w-multiplier
divisor of tp f m -i
(Mann, 1964)
p2i
strictly divides
is strictly divisible by an even power of
n
and this implies (Theorem 2.19) p.
This theorem, which (as shown above) is an almost trivial consequence of the prime ideal structure of
@(~w)
is an extremely important result.
cyclic difference sets are concerned,
For, as far as
it contains both the Bruck-Ryser-Chowla
theorem (odd v) and the theorem of Hall and Ryser.
In order to establish the
dependence of these results on Theorem 2.7 it is convenient to know the connection between diophantine equations of the type
x
2
= oy
2
+ Bz 2
5, ~
and the so-called Hilbert norm residue 'symbol
integers
(~,G)p.
(2.21)
The basic fact is that
28
equation (2.21) has a solution in integers
x, y, z,
(G,~)p = +l
G, ~
for all finite primes
fact is written
(~,~)~ = +l].
p
and
not all zero~ if and only if,
are not both negative
See Jones (1950, p. 26 ff.) for this as well as the
other properties of the Hilbert symbol which are needed.
Corollary 2.8 exists and let
(Yamamoto,
re
has a solution in integers
e
(~,~)p : (~,~)p
(2.23)
(~,~)p(~,T)p : (~,~)p
(2°24)
1963)
x
(p = ~ allowed):
(2.22)
Assume that a
2
v, k, ~ - difference set
for an odd divisor
strictly divides
r
These are
(~,~)p : ± z
q* = (.l)(q-l)/2 q
prime such that
[this last
n,
q
of
v.
Then if
r
it follows that the equation
.(q-l)/2 2 2 + (-1)" - qy = z
x, y, z,
is a
not all zero.
(2.25)
Or, equivalently,
that the
Hilbert norm residue symbol
(2.26)
(re,q*)p = + 1
for all primes
Proof.
p.
Since
assumed to be a prime. x,y,z = l,O,r e/2.
(qlq2)* = q{ q~ If
e
(x,y,z = a,brJ,cr j)
rx
r = q ~ 3(4).
x,y,z = a,b,Co If
q
m a y be
is even, equation (2.25) has a solution
Thus attention may be restricted to odd
case (2°25) has a solution
has a solution
it follows from (2.24) that
r = q ~ 1(4)
2
+ (-1
Now
e = 2j + i.
In this
whenever the equation
~"(q-1)/2qy 2
x,y,z = i,i,0
then since such a
: z
2
(2.27)
is a solution of (2.27) when q
has a representation
(see
Nagell, 1951)
q : s2 + t 2
x,y,z = s,t,q
is a solution of (2.27).
note 2) requires
where
r, q*, -rq*
s
and
t For
are integers, it follows that r ~ q
Legendre's test (section II.B.
to be quadratic residues of
q, r, 1
respectively.
Of these conditions, the last is trivial and the second follows from the first by quadratic reciprocity (Nagell, 1951). residue of
q
(Theorem 2.7)
As for the first, if
then, by Euler's criterion~ e
would be even.
r
were not a quadratic
r (q-l)/2 e -1 (mod q)
and hence
This contradiction establishes the existence of
a non-trivial solution for equation (2.27) and hence for equation (2.25), i.eo, the corollary has been established. In terms of the Hilbert symbol, the theorem of Hall and Ryser becomes (n,W*)p = +l
for all odd divisors
of equations (2.23), (2.24), (2~26).
w
of
v
and this is an immediate consequence
Using this for
w = v, v
odd, yields
+l = (n,V*)p = (n,V)p (n,(-l)(V-l)/2)p
while
k 2 = Zv + n
provides
+l = (n, hV)p = (n,h)p(n,V)p .
Together these equations show that
(n,h)p = (n,V)p = (n,(-l)(V-1)/2)p.
Hence
(n,Z)p (n,(-l)(V-l)/2)p = (n,(-l)(V-1)/2k)p = +l.
Ioeo, the Bruck-Ryser-Chowla condition for odd
v
has been established as a direct
consequence of Theorem 2. 7, In particular, if
v,k~ Z = 241, 16, 1
the Bruck-Ryser-Chowla equation
30
z
2
= 15 x
2
Hall-Ryser However
+ y
2
has a solution
condition
360 m -1
z = y = i, x = O.
is only stronger
modulo
241;
As 241
for composite
v,
is prime and the
it must also be satisfied.
thus Theorem 2. 7 shows that no difference
set
exists. Another
result of Mann (1964)
Theorem 2. 9 .
Let
n
is a square (i)
(ii) (iii)
(iv) (~)
qv
1
be a w-multiplier
w > 1
set for some divisor Then
t
of
v.
23 n = noq
or
modulo
Let
t f E -1
t
for some prime
ql / q
Since
some integer
residue
tI
so is
divisor
and
also divides
2 n = nln 2
2
2 = n1
and
Let
q
n2 = q
k 2 = hv
+ n,
n 2 > 1. n
tl I / -1
modulo
ql
fl"
thus
8(~w l) = ~
O(~w)
for
=
q
So k - ~
n
d
for
w
it follows
2 n = nln 2
w
th
roots of
where every prime
so from now on assume
that
w
v of
were
and
from
and hence n2
w
and
2 n = nln 2 n2
are
and apply the same process
w; dq s
would show
w = qS (s >_ l)
in the field of
or
playing the role of If
(2.28)
m a y be assumed to be square free.
be a prime divisor
necessarily.
then the same process with assumption
n2
the theorem is proved,
above with the odd prime m
w
of
2 n = nI
By Theorem 2.1 it follows
odd.
tf;
then
= n ~wr
ideal factorization that
n2
v
Then
unity (Theorem 2.19) , it follows
necessarily
f.
In this later case
and all integers
is w-multiplier,
From this and the prime
n 2 > i.
for some integer
q
is another prime divisor of
t
r.
of
[8((w)]2
with
q.
w
s > 1
is a quadratic
if
Unless
modulo
v, k, ~ - difference
4
for all multipliers
of
of a non-trivial
is odd
w = qS
Proof.
is:
this yields
n = m q.
with
and
q = i
n = 4q n -= v = 0
2
d > 1
modulo
q q
Thus
(d,q s) = l,
which contradicts with
as
prime.
As
that
q2
the
31
divides
n;
thus with
is not assumed that As before, in
nO = nl/q
the desired form
2 3 n = noq
emerges, where it
(no, q ) = 1. K(~q)
there is an equation analogous to (2°28), namely
[e(~q)] 2 = n ~q
for some integer
~ > 0o
By definition
(2.29)
q* = (-l)(q-l)/2q
Gauss (see Nagell, 1951, section 53) that with
and it was shown by
~q = e 2~i/q
= z+ ~q+ ~q4 + ~q9 + . .. + ~( q-l)2
Hence
~
is an algebraic integer of
automorphism defined by Further, if into
t
whenever
Since
~
q,
t
lit is invariant under the field is a non-zero square modulo
then the mapping
is an algebraic integer of
Y = e(~q)/nl~q~q*
is an element of this field.
=
=
n--~
i.e., since
~,
is not a square modulo
- ~q*. ]
the fraction
~q~
K(~q).
Y
n(-l) (q!l)/2
satisfies the equation
K(~q)
modulo 4o ~
satisfy As
Y
x 2q = i; is a
t ~q
~q~ K(~q)
q.
takes
it follows that
Indeed, since
(2o31)
[ (-l)(q-l)/2
x 4q = i,
an algebraic integer but also a root of unity in in
(2.30)
it follows that it is not only K(~q).
But all roots of unity
thus it follows from equation (2.31) that
2q th root of unity, it is
± ~
for some
q
th
q m i
root of unity
Thus
@(~q) = ~ q nl"~'~q*
and as
is a q-multiplier it follows from congruence 1.14 that
Now if
is not a square modulo
q,
(2.32) s 8(~ ) = ~q
e(~q).
it follows from this equation together with
32
(2.32) that
t
t ~q -~ ~q
since, as noted above, contradiction, so
t
nl~q. = ~(±
maps
(2.33)
~ nl~q~. )
W~q* into
- ~q*.
But (2.33) is a
necessarily is a non-zero square modulo
q.
Of the conclusions of Theorem 2.9 only the last one remains to be verified. If the prime
ql ~ q
were another (necessarily odd) prime divisor of
which there existed a multiplier modulo
ql'
tI
and integer
fl
such that
then, by the process used above, it would follow that
23 n0q
(since it cannot be both
and
23 m0ql).
nor
q
fl tI n
v~
for
~ -i was a square
This contradiction completes the proof
of Theorem 2. 9 . Note that neither
tI = t
Consider the parameters 2
dividing
v,k,h = 813, 29, 1.
nO
is excluded by this theorem.
Theorem 1.1 (section I.E) shows that
is a multiplier, hence also a q-multiplier for
q = 3.
Thus Theorem 2. 9 shows
that no such difference set exists. Before establishing any further existence tests it is convenient to have some results concerning congruence relations in cyclotomic fields°
Let
C
be a number
theoretic function (i.e., a function which is zero except on the integers) and define the difference operator
A(O)
by
a(~) c(i) : c(i + 0) - c(i)
where
p
is a rational number not necessarily an integer.
exists such that
Theorem 2.10 decomposition of
•(n) C(i) = 0
for all
(Yamamoto, 1963) N.
Let
m
Let
integers of the cyclotomic field d
be a divisor of
N
then call
N
K(q)
and let
~
C
~i """ Ps~s
N = Pl
be relatively prime to
number theoretic function with period
let
i,
N,
whose values and let
If an integer
periodic of period
be the prime power let
C(i)
C
be a periodic
are algebraic
f ( x ) = ~i_-1 C(i)x i.
be an integer of
n
K(~m).
Further,
n.
33
Then, in order that
f(~)
m 0 (mod G)
for all divisors
r
of
d,
it is
necessary and sufficient that
tI ts -tl-1 -t -I P! "'" Ps A(NPl ) ... D(Np s s ) C(i) " 0
for all
let
i
and for all
Proof.
(i)
u = 0.
Now
Let
tl,...,t s
such that
s = i, N = p~, d = pU
~-i i P ~-
~i
i=O
j=0
tI ts Pl ''" Ps
(mod J)
d.
divides
and, proceeding by induction on
U,
~i+jp ~-I C(i + jp~-l) ~N
f(~N) =
p~-l_l i
I 1
i=o
j=l
[C(i + JP~-l) - C(i)] ~N+JP~-I
~-l since
~P
~i+jp ~'l N
= ~p for
K(~m) ; thus
is a primitive
pth root of unity.
0 _< i < p~-l, i _< j < p f(~N) - 0 (mod ~)
for all these values of
i
and
~(pZ)
integers
form an integral basis for
if and only if j.
The
C(i + jp~-l) _ C(i)
K(~mN)
over
0 (mod (~)
This condition is equivalent to the desired
one
for all (2)
i. Let
t, 0 < t < u.
Thus the theorem is proved for s = l, u > 0
s = I
and
u = 0.
and assume that the theorem is true for all integers
Let
p
g(x)=p
~-l
I i=O
-1
c(i) x~p
3~ and note that
p
~-i p -i
1
f(x p) i=O
Since
i=O
f({N) m O (mod G)
modulo
G
~l c(i + jp~-l)
C(i)xiP __-
implies [see part (i) above] that
mod(~,x N - I) .
t f({~ ) m 0 (mod ~)
for all
t equivalent to 0 < t < u - i. equivalent to
f(~
) m 0 (mod ~)
and
0 < t < u - i,
for all
t
such that
0 < t < u.
Now assume s.
s > i
where
k
0 < t < u
is
for all
t
such that
and
pt+l h(p~-t-2) C(i) m 0 (mod 5)
that is, to
pt A(p~-t-l) C(i) m 0 (mod ~)
Thus, the theorem has been proved for
s = i.
and assume the validity of the theorem for smaller
~i ~2 ~s N = NIN', N I : Pl ' N' = P2 "'" Ps ' d = dld',
Put
d I =(Nl, d), d' = (N',d).
exist integers
g({~ ) ~ O (mod 5)
pt A(p~-t-l) C(i) ~ 0 (mod G)
such that
values of
such that
By the induction hypothesis these last two congruences are
t
(3)
t
t
for all
and
C(i) ~ C(i + jp~-l)
it follows that
Thus the condition
N1
(mod x N - i) o
j=O
f(x p) ~ g(x)
r = rlr'
x ip
rI
Any divisor
divides
j, k
dI
such that
determined modulo
NI-I f(x) -
N' -i Z
and
r r'
of
d
can be uniquely written as
divides
d'.
i ~ N'j + Nlk (mod N) N'.
For any integer with
j
i
there
determined modulo
Hence
C(N'j + Nlk ) x
N'j+NIk
(mod x N - i)
j:o k=o NI-I r f(~N ) =
Z j=O
N'-I ~ k=O
NI-I Nlr
~N'jr < i k r = C(N'J + Nlk) ~N
c*(~ N j=O
~N'rj
,j)
~N
35
NW_l
where
C*(y,j) =
Z
C(N'j + Nlk)Y k.
k=0
N'
Now
IN
= ~
is a primitive
NI th root of unity and
NI IN = ~
root of unity. Further, the condition
f(~)
the same congruence for all primitive
N th roots of unity.
for some
IN
~ 0 (mod ~)
is a primitive
for some Thus
IN f(~)
N 'th
implies ~ 0 (mod ~)
implies that
NI-I Z
C*(~r' ,j)
~rlJ
(mod ~)
o
j=O th
for all primitive
NI
qo
C*(q
Note that the
(mN',N1) = lo
roots of unity r'
,j)
~
and for all primitive
are algebraic integers of
Thus applying the theorem for
s = 1
N'
K(~N,)
roots of unity
and that
to this case, i.e., to the
polynomial
NI-I
~,
c*(n
rW
,j) x j
j=0
yields the result that
f ( ~ ) ~ 0 (mod ~)
for all divisors
r
of
d
if and only
if
tl -tl-i r' Pl aj(NlPl ) C*(~ ,j) ~ 0
for all ~j(p)
tI
such that
tI Pl
divides
dI
(mod Q)
and for all divisors
(2.34)
r'
of
indicates that the difference operator applies to the argument Congruence 2.34 may be rewritten as
d'o j.
Here
36
N t- I
tI Pl
-tl-1 •j(NlP 1 ) C(N'j + Nlk)n r'k -=0
~
(2.35)
(rood G)
k=0
for all
tI
and
r'
such that
t1 Pl
divides
dI
and
r'
divides
d'.
Apply the
induction hypothesis to the polynomial
N'-I
tl
~
Pl
-tl-1 aj(NlP 1
) C(N'j + Nlk ) x k
k=0
which can be done since
N'
has
s - 1
distinct prime divisors, since the co-
efficients of this polynomial are algebraic integers of (m,N') = 1.
K(q)
and since
Thus congruence 2.35 holds if and only if
t I t2 ts -tl-i -t2-1 -t -i Pl P2 "'" Ps ~j(NlPl ) ~(N'P2 ) ... ~ ( N , P s s ) C(N'~ + ~ k )
ti
= Pl
for all
i
-tl-!
ts
"°" Ps ~(NPl
and for all
-t s -i
) ... &(NPs
ti,...,t s
such that
) C(i) - 0
tI t Pl ... ps s
(mod ~)
divides
d.
That is,
the theorem has been established. Folliwng Mann (1967) note that:
Corollar~ 2.11.
Let
~ > 0
be an integer and let
periodic, number theoretic function of period 0 < C(i) < M
N,
C
be an integer valued,
whose values
C(i)
satisfy
and
N-1
o /
c(i){~ ~ o
(~od ~).
i=0
Then, if
N
is the product of exactly
s
distinct prime powers, it follows that
37
< 2 s'l M .
Proof.
Apply Theorem 2.10 with
m = d = l;
this yields
A(Np[1) .,. a(~p~l) C(i) ~ 0
which must hold for all integers (2.36) must be non-zero. congruence Z C(i)~ C(j)
For some
[For otherwise
for any arbitrary value of
~ 0.]
with
magnitude
i.
Consider
0 ~ C(j) ~ M;
(2.36) at
~
i 0.
(mod ~)
i,
say
i0,
(2.36)
the left side of
Theorem 2.10 could be applied to this and this would contradict the condition The left side contains
exactly half of which have negative
of the left side of (2°36) is bounded by
2s'~,
2s
terms
signs.
ioe.,
Thus the
~ ~ 2s-~
as
was to be shown. Corollary 2.11 will be put to good use in the proof of the next existence test.
But first a lemma and a definition are needed.
Len~na 2.12o K({s) ,
and let
w-i Z a.x i, where the a. are algebraic i=0 1 l (s~w) = (m~w) = 1 for some integer m. If Let
A(x) =
A(~)
for
0 < j < w - i,
then
a. -= 0 l
integers and (2.37) holds only for
Proof.
--- 0
modulo
modulo
m
for all
1 < j _< w - l,
m
i.
(2.37)
If the
a.
l
then
a 0 -= a I ~ -.. =- aw_ I
modulo
Assume that (2.37) holds for
0 < j < w - I,
integers of
m.
i.e., that
are rational
38
1
1
1
ao l
. . . 1
#,
1
al
j
.
+.x +. 1
.
(x
~)(x
~)
o. . (x - <w-1 )
it follows
that
w . (i .
.~)
q,)(1 .
This shows that only primes dividing by Cramer's rule, it follows that
w
. . o
(1
~w - 1 ).
may occur in the determinant above.
a. m 0
modulo
m
for all
So,
i.
I
Suppose the modulo that
m.
+
I _
m
for 0
are rational integers and suppose that
Consider the polynomial
~ + ~w ~ 0
modulo a.
ai
modulo m
[certainly possible since
0 < j < w - 1
modulo
m
B(x) : A(x) + ~F(x),
for all
where
~
(m,w) = 1].
is chosen so Then
and hence (by the first part of this lemma) i,
that is
!
a 0 - a I -= ~o. = aw. 1
as was to be proved°
A(1) = ~ ~ 0
modulo
m
B(4 ) ~0
39
Let
p
w = p wI p
f
be a prime and let
with
~ -i
(P,Wl) = i.
modulo
Wl,
then p
w
2
are self-conjugate modulo
w.
divides
follow from
n~
divides
self-conjugate modulo
exists.
Let
v
and
O(~w l) = n
Theorem 2.13
w
that
w,
divide
for some divisor
w.
then
If all the m
m
m
is said m
in the
is self-conjugate
in this field is fixed under
w >l
m
is self-conjugate modulo
e(~)
~ 0
modulo
m.]
w~
it would
Note that if
it is also self-conjugate modulo any divisor of
(Turyn, 1965)
m2
such that
Thus, if a difference set is under consideration for which
w
e(~)
i.e., let
[From the prime ideal factorization of
then every prime ideal divisor of
complex conjugation. m
f > 0
roots of unity (see Theorem 2.19) it follows that, if
modulo
w,
is said to be self-conjugate modulo m
to be self-conjugate modulo th
strictly divide the integer
If there exists an integer
prime divisors of an integer
w
p
n
Assume a non-trivial
and suppose that
of
v.
If
m > 1
(m~w) = 1
then
m
is
w.
v,k, h-difference set is self-conjugate modulo m < (v/w)o
If
(m,w) > i
then
m _< 2r - 1
where
r
e(4) Let
divides
(2.38)
is the number of distinct prime factors of
Proof. that
(V/W)
Since =- 0
is self-conjugate modulo
modulo
w = VlV 2 m.
m
m
for
where
Further,
1 < j <w
(v2,m) = 1
w
(m,w)o
it follows
(as noted above)
- 1.
and where every prime divisor of
vI
let
w-1
e(x) ~
~
ai x i
mod ( ~
- 1)
i=O
where w = v 2.
0 _< a i _< v/w Here
e(~)
First consider the case
necessarily. ~ 0
modulo
m
for
i < j < w - i
(m,w) = i,
i.e.,
and by Lenzna 2.12 it
also
4O
a.
follows that all the coefficients
are congruent modulo
m.
Let
~
be the
1
smallest
such coefficient°
Then
w-1
2
w-1 a.x~
= ~
i=O
where
0 ~ a i - a ~ v/w
zero, then divides
e(~)
ai - ~
Z
n
i
(a i - ~) x i
i=O
and not all the
for all
2
i=O
and hence
Assume now that
w-1 +
ai - ~
are zero.
[For, if they were all
would also be zero~ a contradiction].
it follows that
(m,w) > i
m ~ a i - ~ ~ v/w
has exactly
r
Since
m
as was to be proved.
distinct prime divisors.
Consider
the polynomial
w-i
~(x) =
where,
as usual,
~v I
v2"l i
Xi =
~ a i~vl i:o
is a primitive
v~±
st
~
~ j--o
b xj
j
root of unity and
J+v2 + o bj = aj ~ i
Since
m
+ aj+v2 ~Vl
is self-conjugate
modulo
~J+(Vl-l)v2 "" + aj+(Vl-l)v2
w
and
vI > 1
j; a
0 < ~ < v 2 - 1. further b j,
b. ~ 0
(mod m)
By Le~na 2o12 this implies that for some
j
since
n ~ 0o
yields
m < 2 r-I (v/w)
"
it follows that
v2 ) ~ 0
for
Vl
bj = 0
modulo
m
for all
Applying Corollary 2.11 to such
41
as was to be proved. This theorem shows, for example, that if (n,v) = lo and
[For if
n = k - k.
p
divides
(n,v)
then
v p
is a prime power then
2
divides
Thus the theorem may be applied with
v,k,Z = 56, ll, 2.
since
k2
m = p, w = v.]
particular no cyclic difference set may have parameters Consider the parameter set
n,
Since
=
Av + n
In
v,k,h = 16, 6, 2. n = 9
is a square, all
previous theorems allow the possibility that a difference set might exist with these parameters. If
But it is ruled out by Theorem 2.13, since
(a,c) = l, a b ~ 1
x (1 ~ x < b)
then
b
Theorem 2.1~. let
ql
modulo
c
and
ax ~ 1
modulo
c
is said to be the order of a modulo
(Yamamoto,
strictly divide
v.
1963)
Let
q = 4t + 3
33 m -1
modulo 28.
for any c,
written
ord c a = b .
be a prime divisor of
Assume that any prime divisor
p
of
n
v
and
satisfies
one of the conditions: (1)
order of
p
modulo
q
(ii)
order of
p
modulo
q~
(iii)
is even is
q~-l(q . 1)/2
p : q.
Then, if there exists a non-trivial
v,k,k - difference set, the diophantine
equation
2 + qy ,
0 -< x ,
has a solution in integers
x, y.
Sn = x
Proof. group of
Let
K(~w)
2
w = q~, e ( ~ )
= ~
0 _< y
< v q -~ ,
_
and let
over the rational field;
is a primitive root modulo
w~
If a prime
~
+
~
takes
satisfies
from Theorem 2.19 that all its prime ideal factors in automorphism even;
s2.
If
p
y
f
(Theorem 2.19) that the p-component of
such that V
in
~
into
where
s
p
are fixed by the modulo
pf = -1 (mod w).
K(~)
s ~w
(ii) or (iii), it follows
K(~w)
satisfies (i) then the order of
thus there exists an integer
_< 2vq -~
be a generator of the Galois
then p
x
w
is also
This implies
is fixed under complex
42
conjugation and thus (from integer
2 G
a.
Thus in
~
K(~)
= n)
all the prime ideal factors of
Hence the principal ideals
where
~
is a unit of
it must be the principal ideal
(~)
K(~w).
(To"2 )
and
T
(pa)
for some
are fixed under
are the same, i.e.,
~ = ~T
0`2
It follows from (2.20) and Theorem 2.20 that
thus must be a root of unity in
K(~),
i.e.,
q = £ 4
for some integer
j
where
E=+I. Let
N : ~(w) = q~-l(q _ 1),
EN/2 = g
then
N/2
is odd since
q ~ 3 (4).
So
and
1 = ~l+~+'"+J-2 = N/2 -w <j(l+~+'''+J-2)
shows that
g
is a w
th
root
is replaced by its shift
of unity;
D + u
then
thus h = T
g = + i.
i-~
(2.39)
If the difference set
is replaced by
Thus there will be no loss of generality in assuming that
~ = + i
provided that
it can be shown that
(i - s2)u e - j
has a solution i - s
2
u.
If
is prime to
q.
q ~ 3
modulo q~
then (2.40) clearly does have a solution since then
When
q = 3,
3~
the 3-component of
3-component of
I - sN
is
equation (2.39)
[i.e.,
j(l - sN)/(l - s 2) ~ 0 modulo 3 ~]
j,
since
s
G 2. n = ~
So
~
~ = + I
i - s
is a primitive root of
which of course guarantees a solution
without loss of generality,
u
is the norm of an integer of w = (-I + ~ - ~ )
the complementary difference set
D*
K(~-q).
has
is 3 ~.
3
and the
Thus 3
for (2.40) in this case alsoo
and thus
/ 2 ;
2
provides that
~
thus
divides So,
is fixed by the automorphism
is an algebraic integer of the quadratic subfield
rational integers and
(2.40)
So
T = a + b~
K(-~q) where
4n = (2a - b) 2 + b2q.
T* = - a - b~,
that
and a, b
are
[Note that
~ = a - b - b~
43
and that
~* = b - a + b~.]
Thus by using
~
or by replacing the difference
by its complement or by doing both it can be assumed that
a > 0
and
set
b > 0.
From (2.30) it follows that
l W
i ~(i) ~q = ±
=
i=l
Here
~(i) = i
or
q, {q = e 27~i/q
if
- Zi= ,~-i 0 B(i)
j xi
according ±
~
i
is a quadratic residue or nonresidue
depends upon the value of residue of
x w - i,
w-i f(x) =
as
is a quadratic modulo
yields
x = ~;
~(q~-l)
q
j
and
~
in the relation
otherwise)°
Let
. Z-I
B(i)x i - \ a + b
~(j)x 3q
:
j=l
for all
io
C(i)x i i:0
so apply Theorem 2.10 with
C(i) = 0
of
then the polynomial
i:O
is zero for
iq~_ I ,(i) ~w
i=l
and the sign
~-i ~qw = ~qJ (+ e(~)
0
~i
s = d = i
In particular,
let
and
~ = 0.
This
i = 0, q ~ - l , . . . , q ~ _ q ~ - i
then
B(O) - a = B ( q ~ - l r ) T b ~ ( r )
and since
4(1) = i, ~(q - i) = 0
B(-q~-l).
As
a ~ 0, b ~ 0
Applying this same process to
and
it follows that 0 ~ B(i) ~ vq -~
~
Combining these results establishes Yamamoto
= b - a + be
B(0) - a = B(q ~-I) 7 b = this implies that
yields
the theorem with
shows further that when
v =w = q
r = l,...,q - i
O ~ a , b ~ v q -I.
la - b I ~ vq "~.
x = 12a - b I
and
the only non-trivial
y = b. difference
sets which satisfy the conditions of Theorem 2.14 are equivalent to the set consisting of the quadratic residues of in partieular~
q
or to its complementing
Theorem 2.14 shows that no difference
set
D*.
D Thus~
set exists with parameters
44
v,k,k = 239, 35, 5
(a parameter set not ruled out by the previous results of this
section).
Theorem 2.15. prime divisors of where
~
(Yamamoto, 1963) v
and let
q~
Let
rm
denotes Euler's function.
q = 4t + 3
and
r
v
with
strictly divide
be distinct odd ($(q~)~ ~(rm)) = 2,
Assume that any prime divisor
p
of
n
modulo
r
satisfies one of the conditions:
(i)
Ordqp ~ 0 (mod 2)
(ii)
order of
p
and
modulo
q~
OrdrP ~ 0 (mod 2) ~ 0(mod 4), is
½ ~(q~)
and order of
p
m
is
~(rm), (iii)
p = q
and order of
p
Then, if there exists a non-trivial
4n = x 2 + qy 2 ,
0 ~ x,
has a solution in integers
modulo
rm
is
$(rm).
v,k,h - difference set, the equation
0 ~ y ~ 2vq-~r-m,
x + y
4vq-~r-m
x, y.
Note that, by Theorem 2.7, these hypothesis are only satisfied if square.
is a
Thus as a non-existence test, applied after Theorem 2.7, it can be
restricted to those cases. for parameters
21, 5, 1.
The integers With
v,k,h = 306, 61, 12
P = 7, q~ = 9, rm = 17
3, 6, 7, 12, 14
q = 7, r = 3
(ii) and the equation has solution meters
n
the prime
x = 4, y = 0.
form a difference set 2
satisfies condition
On the other hand, the para-
are not associated with any difference set since
satisfy (ii) of Theorem 2.15 and the diophantine equation
has no solution.
Proof.
Let
group of
K(~)
p
~
where
[respectively K(~R)]
Q = q~, R = rm, w = q~rTM
and
G ( ~ ) = Vo
Note that the Galois
over the rational field is generated by two automorphisms
Krespectively K(~Q)]
p]
fixes every element of the field
and generates the Galois group of
over the rationals.
then there exists an integer
If f
p
K(~Q)
is a prime divisor of
such that
pf ~ -i
modulo
n
~
and
K(~R) [respectively
which satisfies (i), w;
thus (as in the
45
previous proof) the p-component of the ideal for some integer (~)
is
(pb)
Finally, if of
p
a°
Similarly, if
for some integer p
b
is the principal ideal
(pa)
satisfies (iii) then the p-component of
(a consequence of Theorems 2.7 and 2.19).
satisfies (ii) it follows from ((P(q~), $(rm)) = 2
modulo
divisors of
p
(~)
that the order
w
is
½ q0(w); thus there are two (Theorem 2.19) prime ideal
(p)
in
K(~w)
quadratic subfield
and since
OrdQ p = ½ (P(q~) they originate in the
K( ~-q ).
Hence the ideal
(~)
is fixed by
p,
fixed by
o2
and originates in
i_~ 2
K(~-q)
.
Thus V
= ~, 1 - p
implies (by 2.20) that unity in
K(~).
of unity.
Now
o2,
if
= 55 = 1
l-p
is an
,i
and
q = 3
5 = e ~R
then
and
5
K(~-q)
are units in
1-p
K(~);
9, 5
of unity and
hence
is a
q~th root
= 81-o~ = + 1.
Thus
~
of degree
2~(r m)
i, j
with
g,g' = + i a.
q = 3(h) has no units except
D + ~r TM
but allows
5
~ = + 1.
+i
unless
[Shifting
5. ] Shifting again by a multiple of
to take the shape
g'
g i ~3a
or
for
unless
Thus
q = 3;
[This last follows
D
q~
q = 3. ]
E = +l
for a judicious choice of
assume, without loss of generality, that has no effect on
is
over the rationals.
By the same proof as above, see equation (2.39), it follows that
m r
are roots of
51"~2
may occur for some integer
that by considering the shift
this
is a root of unity in the subfield fixed by
for some integers
8 = g §3~R
from the fact that
rmth root
K(~R, ~-q )
t ~a~i
~, B
and that (Theorem 2.20)
T(1-c2)(l'P);
K(~Q)
i.e., in the field
= g~
~
But both are
a root of unity of
= 5 where
~
and
one may
by a multiple of does not disturb
9
q = 3.
l.C 2
First consider the case an integer of fixed by belongs to
p,
K(~R, ~-q ) i.e.,
T2
K( ~r*, ~Q)
q ~ 3.
Since
and since
1-p
is an integer of where
K(J~-q) S,
T
= g, = + l K( ~
).
over
a
T in
K(~)
it follows that
it follows that Also
T l-p2 = l,
as before.
~ = K(~-q, ~r*).
then
In this case, for some ifiteger polynomial satisfied by
= ~ = 1
r* = (-l)(r-1)/2r
integer of the biquadratic field does not belong to
T
If
g' = -1
necessarily generates K( ~
), x
2
That is,
- a = 0
[i.e. ~
over
T
2 so ~
is
is T is an
if K( ~
).
is the irreducible
and thus [Mann (1955) Chapter 12] the
46
relative different of (r,n) = 1
T
~
is prime to
algebraic integer of
2 T '
K(~-q)
is
27.
q = 3o
r
is odd,
@
: n
belongs to
T
K({R; ~
K(~)
r;
a contradiction.
satisfies ).
Since
= K(~3)
[here
T
= I
extension
~q of
a subfield of
K( ~-3 ).
K(~R, ~ - ~ )
Now
a ---0 ( 3 ) ]
r
~' = 1
and
7
thus its relative different is
T~
x3 - b = 0
is cubic over
Thus
T2
determines a cubic
K( ~
).
~
a - 0 (3)
and
r
since
On the other hand
72
b
of
K( ~
);
(r,n) = I
and
belongs to
q ~ 3
Thus in every case
T = e(~w)
theorem~ this implies that [with according as modulo
e
K( ~
and, since
3
plays ~' = i
).
is an integer of T = a + b~
).
K( ~'q ).
w = Zq-ll ~(e)~q,
is a quadratic residue or nonresidue of
As in the previous ~(e) = i
or
0
q~ 8(x) - Z iw-i = 0 B(i) x i
xw - 1 ]
f(x)
=
w-l~ , ~ /, Bki)x i -
{' \ a + b
i=O
has a zero at that
K( ~
[see
is
since
72
K({3)
no essential role in that argument, it may be invoked here to show that is an algebraic integer of
2
9
for some integer
which is prime to
Thus
and thus
72
So this case has been reduced to the conditions for
7
is an
it follows that either
divides the diseriminant of
374,
Contradiction.
it follows that
or
properly containing
satisfies an irreducible equation
r ~ 3 = q.
Thus
T 2(l-P) = {3 2a
van der Waerden (1949) p. 171 for this fact].
and that
and
K( ~q-q).
Since
belongs to
since
Since
is implied by conditions (i)~ (ii)~ (iii)~ it follows that the
discriminant of
Let
over
x = ~.
~±
~r(e)
m ~-i x er q
e=:l
for all
io
=
w-1
Ci[ i ")"x
i=l
This implies (Theorem 2.10 with
A(wq -I) ~(wr -I) C(i) -- 0
)
s = 2, d = i
and
~ = 0)
That is,
c(i) - c(i + q~-lm) ~ c(i + q~r m-l) - C(i + q~m-Z
+ q~-lT)
(2.41)
47
for all
i.
Combining equations
B(0) - a - B(q~-!rmj)
for
Now
j = l,...,q - io
i = 0, q~'irm'...,(q- 2)q~-irm
yields
% b~(j) = B(q~r m-l) - B (q~r m-I + jq~-irm)
When
j = I
and
j = q - 1
equation
(2.42)
(2.42) becomes
B(0) - a - B(q~-ir m) W b = B(qlr m-l) - B(q~r m-I + q~-irm)
(2.43)
B(0) - a - B(-q~-l~ TM)
(2.44)
0 < B(i) _< v/w
Ib 1 _< 2v/w
(2.41) for
for all
i.
= B(q~ m-l) - B(q~ m-I - q~-irm).
Thus equation
follows from the same reasoning
from subtracting
(2.44) from (2.43).
(2.4~;) shows that
Consideration
the equation
and
applied to the equation which results
resulting equation analogous to (2.44) produces x = 12a - bl, y = Ibl
lal _< 2v/w
of
~* = b - a + b~
Ib - a I < 2v/w.
4n = x 2 + qy2
and the
Thus with
has a solution of the proper
type and the theorem has been proved. Yamamoto notes that when
Im v = w = q r
the only difference
all the conditions
of this theorem have parameters
are all equivalent
to the set
Theorem 2.16. divisors of
v,
(Yamamoto,
let
q
1963)
Let
q = 4t + 3
be a quadratic non-residue
where
divisor
satisfies one of the conditions:
(i) (ii)
of
n
v,k, h = 21, 5, i;
thus they
[3,6,7,12,14}.
(q0(q2),~0(rm)) = 2 p
sets satisfying
q~ , rm
strictly divide
and
of v.
r
r = 4s + I
be prime
and let
Assume that any prime
ordqp ---0 (mod 2) and ordr(P) ~ O(mod 2) ~ O(mod 4) order of
p
modulo
q~
is
@(q~)
and order of
p
modulo
rTM
~(rm). Then, if there exists a non-trivial
4n = x
2
+ qry 2,
0 < x,
v,k,~ - difference
0 < y < 2vq-~r -m,
se%
the equation
x + y < 4vq-~r -m
is
48
has a solution in integers
x, y.
Again, as a non-existence test, applied afte9 Theorem 2.7, this can be restricted to those cases where
n
is a square.
The integers
0,1,2,4,5,8,10
form a difference set with parameters
v,k,k = 15, 7, 3.
satisfies (ii) for
and the equation is satisfied by
q = 3
and
r = 5
v,k,h = 286, 96 , 32
On the other hand, the parameter set tes%
since
q = ll, r = 13, p = 2
Here the prime
2 x = y = io
is ruled out by this
satisfies (ii) and no solution to the
diophantine equation exists.
Proof. e(~)
= 7
field,
and let
~
%
p
generate the Galois group of
generating the Galois group of
generates the Galois group of of
n
Q = q~, R = rm, w = q~rm,
As in the proof of Theorem 2.15 above, let
K(~R)
K(~Q)
and fixes
K(~)
over the rational
and fixing K(~Q).
K(~R)
If
p
while
p
is a prime divisor
which satisfies (i) then it follows, exactly as in Theorem 2.15, that the p-
component of the ideal While if
p
(7)
is the principle ideal
satisfies (ii) it follows from
(pa)
for some integer
(~(q~),C0(rm)) = 2
that
ord
a. p = ½q0(w);
w
thus there are two (Theorem 2.19) prime ideal divisors of since
OrdQ p = ~(q~), o r ~
TM)
p = £0(r
(p)
they do not arise in
in
K(~w)
and
K(~Q)
or in
K(~R).
Since [see Mann (1955), Chapter 13] they must arise in a quadratic subfield of K(~w)
it can only be Hence the ideal
1-~p
= N
K( ~ (T)
is a unit of
= + 1.
D
Since
the field fixed by
is fixed by K(~w)
unity in this field, i.e., difference set
),
~ = + ~J
for some integer
by a suitable shift (e(q~),~(rm)) = 2,
2
and
O2
K( ~
).
Thus
D + ~q~ + ~'rm
both ~0
j.
~2
and so
and
p
By replacing the
it may be assumed that
2
are powers of
~ 1-°2 = ~yl-p2 = + 1.
~p. Thus
In ~
is
and as such lies in the biquadratic field
= K( ~r, ~-q). Furthermore, ~p Suppose T = -T, then since
2 ~/ ~
is fixed by
~p
is an integer of
1,(1 + $r)/2, (1 + ~q-q)/2, (1 + ~r + ~ it follows that
and originates in
and by (2.20) and Theorem 2.20 it must be a root of
fact they are both even powers of fixed by both
~p
~p.
~ = (c ~-q + d ~fr )/2
+ -~qr)/4
and thus belongs to ~
K(-~qr).
and since
form an integral basis for
for some rational integers
c, d
0,
such that
49
c =- d
modulo
2.
Thus
4n = 4 ~
=
c2q + d2r,
and since
equation
solution in integers
not all zero.
x,y,z = 2 ~ n ,
c, d
II.B note 2) shows that this only happens when which contradicts
the hypothesis.
Thus
x
2
square, this implies that the diophantine
q
n
- qy
2
is necessarily - rz
2
= 0
But Legendre's
a
has a test (see
is a quadratic residue of
7 ~p = 7
i.e.,
7
is an algebraic
r, integer
K( ~ ) .
of
So
7 = a + b~
is the Gaussian
+ 1
: 0
or if
i
for some rational integers s qr-1 . i -i=l ~(1)~qr"
sum
/2
where
if
~
Here
if
i
is a quadratic non-residue of
m
w-i ~
f(x) =
f(x)
= 0;
(2.30)].
Let
~ 2B(i)x i - \ 2 a
2B(O) - 2a - 2 B ( j q # - i r
for
j = l,j'
q
TM) +
j : i,
(j'
for
i
yields
~*
+ qry 2
la-bl
i.
modulo
~-i m-i r
and
r
[This of
xw - 1
=
~r, and let
C(i)x i
s = 2, d = i,
- 2)q~-lr TM
~ b - 2 B ( q # r m-1 + j q # - i r m )
]a I _< 2v/w. of
q)
Further,
_( 2v/w.
q
otherwise.
That is, equation
i =0, q~-lrm,...,(q
alsoo
that is
i=0
for all
a quadratic non-residue
associated with 2
1951);
representations
Apply Theorem 2.10 with
it follows that
Ibl _< 2v/w
4n = x
r, = -i
~(j)xjq
b = 2B(q#r m - l )
Theorem 2.15, that
equation
and
8(x) ~ Zi= lw-1 B(i) x i
&(wq -1) •(wr -1) C(i) = 0
for
q
+ 2b
x = ~.
and combining these equations
From which,
~(i) --
j=l
has a zero at
thus
and
of the analoguous
i=0
then
2wi/qr
~ = (-1 + -~qr)/2
is a quadratic residue of both
of both
is a consequence
given by equation
~qr = e
where
is the Jacobi symbol (see Nagell,
(i,qr) > 0, = I
representation
a, b
yields
_+ 2b @(q + j r )
.
Combining these equations
provides,
as in the proof of
the analogous equation
Thus with
(2.41) holds
x = 12a-bl,
(for
y = Ibl
j = i) the
has a solution of the proper type and the theorem has been
5O
proved. Yamamoto notes that in this theorem when q, r
are twin primes (i.e., either
v = w
q = r + 2
or
then in fact r = q + 2)
v = qr
where
and the only non-
trivial cyclic difference sets which occur are the so-called twin prime difference sets;
these have parameters
v,k,h
q r , ( v - 1)/2,
=
(v-3)/4.
These difference sets
are discussed further in section V.D. below. Let
p
n = k - h divides so
p
be a prime divisor of both it follows that
k
r
the relation
divides
n
p
v
divides
and
k
and
k ( k - i) = h ( v - i )
also.
Suppose
p
r+s
n, h
then since also.
pr
implies
pS
strictly divides
v
(ii)
r < s
implies
pr
strictly divides
v.
the p-component of
v,
pr
strict ~
is the p-component of
r > s
p O.
it follows that
h;
Then
h
all the coefficients of
this congruence provides Since
R(x)
are non-negative alsoo
k 2 = noR(1 ) + hv;
thus
x = 1
R(1) = n 1.
(t,v) = l,
e(x)6(x -1) m e(xt)e(x -t) ~ n0n I + ZT(x)
thus
Further with
e(xt)e(x-t)e(x)e(x "l) e [n0n I + ZT(x)] 2,
As
(mod x v -
R(x)T(x) m R(1)T(x) m niT(x )
comparison of these two results yields
x v - l)
while (3.7) gives
[n0R(x ) + %T(x)] [n0R(x-1 ) + %T(x)]
for this same product.
(mod
modulo
l)
x
v
- ±,
a
58
R(x)R(x -I) ~ n~
This implies
[since
R(x)
(mod
x v - 1).
has non-negative coefficients] that
single non-zero term, i.e.,
R(x) m nlx's (mod
x v - i)
R(x)
has only a
for some integer
s.
Thus
congruence 3.7 implies
e(x-l)e(x t) ~ n x s + ~T(x)
Multiplying this last congruence by
e(x)
(mod
and simplifying yields
o(x t) ~ xSe<x)
i.e.,
t
x v - 1).
(mod x v - l)
is a multiplier of this difference set as was to be proved.
This theorem was first established for cyclic projective planes
(i.e.,
h = 1
only) by Hall (1947) and later extended (Hall and Ryser, 1951) to the case of general
h
in the form stated in I.E. above.
generalization. known case.
The conditions
nO > h
and
Theorem 3.1 represents a further (n0~v) = 1
are superfluous in every
For~ no cyclic difference sets are known with
prime divisor
p
of
n
(n,v) > 1
and every
is a multiplier of every known cyclic difference set.
Morris Newman (1963) extended this result slightly by showing that the odd prime
p
is always a multiplier whenever
n = 2p
and
(7p,v) = 1.
Turyn (1964)
generalized Newman's result.
Theorem 3.1A
(Turyu)
that for every prime
p
Let
dividing
n = 2n 0 nO
t
nO
odd and prime to
there is an integer
pJP -= t
Then
with
modulo
jp
v.
Suppose
such that
v .
is a multiplier of every difference set with these parameters, provided
merely that
t
is a quadratic residue of
7
whenever
7
is a divisor of
v.
59
(7,v) = i
Note that in particular this eliminates the condition result.
For if
residue of
7
n = 2p a (a
odd), the assumption that
implies that
contradiction that
a
t3 ~ p3jp ~ -i
(mod 7).
t ~ pop
from Newman's
is not a quadratic
So Theorem 2.7 provides the
is necessarily even.
Mann and Zaremba (1969) investigate the situation when is not a quadratic residue of
7.
7
divides
v
and
However they do not resolve it completely.
particular, they find no case where
t
t In
is not a multiplier.
On occasion it is possible to establish the existence of a w-multiplier for some divisor multiplier.
w
of
v
Frequently this is of importance in constructive existence tests.
[Section III.C.
contains an example of such a case.]
under the conditions that
t
even when it is not possible to show the existence of a
(n0,w) = l, n O > hv/w, pJP ~ t
is a w-multiplier.
However the condition
thus this is not of much use.
Theorem 3.2° n
Let
w
there is an integer
jp
By retracing the proof above (mod w)
n O > hv/w
it can be established is rarely satisfied;
The following related result is more useful:
divide
v
and suppose that for every prime
p
dividing
such that
Jp p
with
(t,w) = i.
parameters
Then
t
~ t
(modulo
w)
is a w-multiplier of every difference set with these
n, v.
Of course the proof of this theorem is almost identical to that of Theorem 3.1 above.
The congruence analogous to (3.7) above, being
9(x)~(x -t) =- nR(x) +
hv
(i + x + ..- + x w-l)
(rood
x w - i)
(3.8)
W
with above.
R(1) = 1.
R(x)R(x -I) ~ 1 (rood xw - i)
But this implies that
must be that 3.1.
From which
R(x) = x -s
R(x) = + x "s
for some
s
is deduced in the manner and since
R(1) = l,
it
From which point on the proof concludes as for Theorem
6O
A surprisingly important fact about cyclic difference sets is that:
The?rem 3.3.
Minus one is never a multiplier of a non-trivial cyclic
difference set. This fact was known for several years prior to any publication of its proof. This accounts for the anomaly that it is often referred to in publications which predate the papers
[Johnsen (1964), Brualdi (1965) and Yates (1967)] containing
proofs.
Proof.
The result (proved in section III.B. below) that any multiplier o f a
difference set Assume that
D
-1
belongs to
D
x - y ~ d
is a multiplier and that only if
and
tions of
d
fixes at least one shift
-x
(= v - x)
(-y) - (-x) ~ d
unless
x = -y.
for
Hence,
D + s D
is fixed by
does also. x, y if
in
D;
d ~ 2x
0 ~ x ~ v/2. of Thus
D
for some
So
d
x
in
for every
x
in
D
y ~ x (y
(0 ~ x ~ v/2)
h
in
D)
v/2
v.
is represented at least
in
D
(such
d
is necessarily even. occurs provided
Whence
y ~ x - v/2 k - i
y - (-y) ~ d
also.
x ~ y + v/2.
Thus,
also belongs to
times.
Hence
D.
h ~ k - i
So, only trivial cyclic difference sets have
as was to be shown.
has been shown (Mann~ 1952) that
B.
x
occurs an even n~nber
such that
It is, of course, not necessary that a multiplier
set only when
d
x - (-x) ~ d
the element
and the difference set is trivial. -1
for some
then
Thus
which can only happen for even
Thus the difference
multiplier
then
x
can appear an even number of times as a difference of elements
only if there exists a 2x m 2y,
D,
D.
i.e., that
they are distinct representa-
0 # d ~2x
of times as the difference of elements of
-1,
Consider the differences
must exist if the difference set is non-trivial),
If
of the difference set is used.
n
2
t
divide
n.
However,
is a multiplier of a non-trivial difference
is even.
Difference Sets Fixed by a Multiplier The use of a multiplier
t
for constructing a difference set is greatly
it
61
facilitated by the assumption that the set is fixed by the multiplier (i.eo, that tD ~ D modulo v).
As mentioned in section I.E. every multiplier
set determines an automorphism of the associated block design. the incidence matrix of the block design, then the multiplier tation matrices
P
and
Q
(Q
takes coltunn
x
t
That is, if t
into column
of a difference A
is
determines permu-
tx
modulo
v)
such
that
PAQ = A
since (see section I.C.)
A
and thus
(3.9)
A-1pA = Q-1
Hence (by
is non-singular for non-trivial designs.
well known facts of linear algebra)
Tr(P) = T r ( A - % A )
where
Tr(X)
elements).
denotes the trace of the matrix But
Tr(P)
Theorem 3.4.
fixed by
then
t
[Similarly, if
is the number of objects fixed by the
tx ~ x
modulo
also fixes the t
D
v,
which is
d
shifts
E + j(v/d),
shifts of
is one such shift, then all others are of the form
D
t,
w
modulo
E
for of w.
E + J(w/8)
Hence
there exists
In fact if
is a w-multiplier for some divisor (t - 1,w) = 8
(t - 1,v) = d.
with multiplier
shifts fixed by the multiplier.
necessarily fixes exactly w
Tr(Q)
Given a difference set
(t - 1,v) = d t,
(i.e., the sum of its diagonal
So, by equation (3.10), the number of shifts fixed by the multiplier
is the number of solutions of
exactly
X
is the number of blocks (or shifts of the difference set)
fixed by the multiplier, whereas multiplier.
(3.1o)
= Tr(Q -i) = Tr(Q)
is a shift
j = 0,1,...,d- 1. v
then If
t
E
modulo
modulo
w,
for
j = 0,1,...,8 - i.] Suppose suppose that
tl, t 2 D
is also fixed by
Theorem 3.5.
are both multipliers of the same difference set
is fixed by t 1.
If
t !.
Then
tl(t2D ) = t2(tlD ) = t2D;
D
and
that is,
t2D
So
t l, t 2
are multipliers of the same difference set then
t2
62
permutes the shifts fixed by
t I.
Thus if any mu!tiplier
t
fixes only one shift of the difference set then
that shift is fixed by all multipliers.
Theorem 3.6.
(i)
Hence
If there exists a multiplier
t
such that
(t - l,v) = i,
then exactly one shift of the difference set is fixed by al__!multipiiers. (k,v) = i
(2)
If
then there exists at least one shift fixed by all multipliers.
Part (2) follows because difference set has
(k,v) = i
insures that exactly one shift
e I + e 2 + --- + e k m 0 (mod v).
E
of the
Such a shift (being unique) is
certainly fixed by any multiplier. Some additional fixed shift results which apply only to the case contained in section IV.A.
h = 1
are
The fact that every multiplier fixes at least one shift
of the difference set was shown by MeFarland and Mann (1965).
Part (2) of Theorem
5.6 is due to J. Jans and the remainder of this section restates for arbitrary
h
results of Hall (1947).
C.
Multipliers and Diophantine Equations Suppose it is known (by Theorem 3.1 or otherwise) that a hypothetical
(Theorem
difference set has a multiplier
t.
Then
set is fixed by the multiplier.
Thus, there is no loss of generality in assuming
that the difference set is a union of sets modulo v.
3.4) some shift of this difference
{a,ta, ...,tra-la],
where
tma -= a
So
Lemma 3.7. union of sets
If a
v, k, ~
[a, ta, .... tm'la),
difference set with multiplier where
tma ~ a (modulo v),
distinct elements and forms a difference set with parameters The sets
[a,ta,...,tm-la)
t
exists, some
has exactly
k
v, k, ~,.
are often called blocks fixed by the multiplier
t.
The number of distinct elements in each of these fixed blocks is always a divisor of the order of
t
modulo
v (this follows, for example, from Theorem 60 of Nage!l,
1951) and is in fact always equal to that order unless prime all the blocks (save
[0])
(a,v) > i.
Thus when
v
fixed by a multiplier contain the same number of
is
63 distinct elements
m
( = the order of
set can exist only if
k = jm
or
t
modulo
jm + i,
v).
In this case a difference
for some integer
j.
Lemma 3.7 is not always easy to apply (for sometimes there are many unions of these blocks having cant information
k
almost trivially.
By Theorem 3.1 above, 151 is prime and
k = 51
with
6
14]
and
{0},
modulo 151)
or
{3, 6, 12],
so a difference
modulo
[9, 15, 18],
set candidates
These are equivalent
15,
Now set
contradic-
that
2
{7, 14] are
is a
and two
{3, 6, 7, 12,
(see section I.B. for this
sets, as is easily verified.
set exists,
is a multiplier.
Theorem 3.1 establishes
only non-trivial one known in which every residue If a difference
15; i
it provides signifiv, k, ~ = 151, 51, 17.
151]
is
0
Thus the only difference
(7, 9, 14, 15, 18]. difference
modulo
21, 5, i,
the fixed blocks are
elements each.
definition)
76
were congruent to
For the parameter values
multiplier~
Consider the parameters
t = 76 [ ~ 214 ~ (17) 35
m ( = order of
could only exist if tion.
distinct elements) however frequently
[This difference d. l
has a c o ~ o n
then its Hall-polynomial
set is the
factor with
9(x) = x
dl
+ -'. + x
v.]
¢~
satisfies the congruence
e(x)e(x -1) ~ n
Thus, for any divisor
w
+ ~(1 + x
of
v,
+ ... + xV-1)
(mod
xv-
1).
(3.11)
one has
e(x) ~b o +bzX+... +bw_ZZ - I
(mod Z - Z )
(3.12)
e(x)e(x -I) ~ n +
(mod
(3.13)
where
bi( _< v/w)
yields
(comparing
Lemma 3.8.
~v
(i + x + .-. + x w-l)
is the number of coefficients
in
in congruence
If a difference
there exists integers
dj
set exists,
b i (i = 0, ...,w - l)
D
satisfying
x w - i)
dj m i nod w.
This
3.13).
then, for every divisor
w
satisfying the diophantine
of
v,
equations
64
w-i
w-i
Z
b. -- k~ l
b.2= n + i
hv -~'
(3o14)
0 < b, < v/w
i=0
i=0
and
w-i
~, b.b..= i l-j
(3.15)
;~v/~
i=O
for
j = l,...,w-
i.
(Here the subscript
i - j
is taken modulo
w.)
An example of the application of Lemma 3.8 is provided by the parameters v, k, h = 70, 24, 8. thus also for
Here
w = 5, 7.
2
is a w-multiplier
Consider a shift of
with
O(x)
fixed under the multiplier
2.
The residues modulo
impart certain restrictions
on
O(x)
modulo
Mod 35
0
w = 35 modulo 35
x5 - i
(Theorem 3.2) and x 35 - i
which is
break into fixed sets which
and modulo
x 7 - 1.
Mod 7
Mod 5
0
0
1,2,4,8,16,32,29,23,11,22,9,18
4(1,2,4)
3(1,2,3,4)
3,6,12,24,13,26,17,34,33,31,27,19
4(3,6,5)
3(1,2,3,4)
5,10,20
3, 6, 5
3(0)
15,3o,25
1,2,4
3(o)
7, 14, 28, 21
4(0)
1,2,3,4
Thus
0(x)
~ o o + Cl(X + x 2 + ... + x 18) + c3(x3 + ... + X 19) + C5(X5 + x I0 + x 20)
+ c15(x15 + x 30 + x 25) + c7(x7 + xl 4 + x 28 + x 21)
(mod
x 3 5 - i)
65
where
c. (0 < c, < 2)
where
is the coefficient of
(mod
x 7 - i)
e(x) ~ b 0 + bl(X + x 2 + ~
(mod
x 5 - l)
and
+ x 4)
0 ~ b. ~ 14.
b 0 + 4b I = 24, b 02 + 4b~ = ]28.
Then
For
equation (3.14) becomes
has only two solutions with a0 = 6
c I = c 5 = O. x~x
3
and hence If
w = 5
equation (3.14) becomes
b 0 = 8, b I = 4
implies (from the residue table above) that
Thus
Further
e(x) ~ a 0 + al(x + x 2 + x 4) + a3(x3 + x 6 + x 5)
0 ~ a. ~ i0
w = 7
x i.
c7 = 1
is the unique solution. and hence that
a 0 ~ 4;
these are
a0 ~ 4 .
a0, al, a 3 = 6, 2, 4 then
For
2 = 96 + a 3)
a 0 + 3(a I + a3) = 24, a~ + 3(a
This
c O = 2.
When
a I = 2, a 3 = 4
a I = 4, a 3 = 2
then
c 5 = 2, c I = I, c 3 = c15 = 0.
or
which 6, 4, 2.
c15 = 2, c 3 = i, Since
transforms this second solution into the first (and takes the difference
set into an equivalent one) it is only necessary to consider one solution. without loss of generality, modulo
Thus,
x 35 - i
e(x) ~ 2(1 + x 5 + x I0 + x 20) + x + x 2 + x 4 + x 7 + x 8 + x 9 + x II + x 14
(3.i6) + x 16 + x 18 + x 21 + x 22 + x 23 + x28 + x 29 + x 32 •
It can be seen (after some searching) that no difference set polynomial satisfy congruence 3.16. 24, 8
D.
Thus no difference set with parameters
@(x)
can
v, k, h = 70,
exists.
Polynomial Congruences If a difference set exists then, for every divisor
exist polynomials
ew(X), G[w](X )
o(x) ~ e[wl(X)
w
of
v~
there must
with rational integral coefficients such that
modulo
xw - l
(3.17)
66
O(x) ~ 8w(X )
where
fw(X)
fw(x)
(3.18)
is the irreducible polynomial satisfied by the primitive
of unity over the rational field. non-negative. [@w(X),
modulo
Furthermore,
the coefficients of
w
th
roots
O[w](X )
are
Conversely (by the Chinese Remainder Theorem), the set of polynomials
all w's dividing
8(X) -
v]
uniquely determines
V
e(x)
8w(X ) BV, w(X )
(mod
modulo
x v - i.
x v - i)
In fact,
(3.19)
wlv
where
V r
rlw
and
b
is the MSbius function.
S[w](X) = ~
r
X
- i
Similarly
Sd(X) Bw, d(X)
(rood ~ - l)
(3.21)
alw
Proof.
By the Chinese Remainder Theorem, congruence 3.19 can be established
by merely verifying the conditions imposed by congruence 3.18.
That is, by
verifying that
v
@w(X) Bv, w(X ) - Or(X )
wlv
for an arbitrary divisor
r
of
v.
modulo
fr(X)
(3.22)
67
[To facilitate this as well as for future reference, recall that if g(x), h(x)
are integral polynomials and
field, then
g(x) m h(x)
any root of
f(x).
~
f(x)
is irreducible over the rational
if and only if
g(r) = h(r)
where
r
is
Further recall that
xm- 1 =
where
modulo
f(x)
f(x),
I~ fd(X) dlm
a~d
fd(~) =
[~ ~jd
(xh- l) ~(d/h)
(3.23)
is the Mobius function.]
Now if
r
and
s
m x x
I - i
s
are divisors of
m/s
modulo
0
modulo
then
fr(X)
when
r
fr(X)
otherwise
divides
s
=
- 1
This follows from (3.23) for the left side at
e2wi/r
when
s Bm'd(X)-
m,
r]s,
r's not dividing r
s sm
]d
divides
_ =
so
m
s
and can be seen by evaluating
Thus, when
d
divides
~ r]s,
modulo
m
fr ( x )
Id
fd(x) Bm'd(X) ~ I
m
modulo
0
modulo
fr(x)
for
r /d
by the standard property of the ~6bius function.
From this congruence 3.22 is
immediate and so congruence 3.19 is established.
For future use~ note that when
d
divides
m
68 m
x -i
where
(')
[x f~(x)]
denotes derivation with respect to
verified by substituting the various gruence 3.24. unity;
(mod
m
th
x.
xm - l )
This congruence may be
roots of unity for
The only difficulty arises when
(3.25)
x
x
and using con-
is a primitive
d th root of
but l'Hospital's rule then shows that the right side is indeed
m
as
desired. A constructive existence test procedure can be based on congruence 3.21. this congruence is used to construct integral polynomials modulo
fd(x).
~d(X)
it requires the knowledge of
for all divisors
d
of
w
such that
O(x) m ed(X)
Neglecting for the moment the problem of finding an exhaustive
list of candidates for [G d}
~[w](X),
If
(i.e., one
ed
Od(X),
note that given a set of integral polynomials
for each divisor
d
of
w)
there is no guarantee that
the polynomial computed from them by congruence 3.21 will have integral coefficients, much less that these coefficients will be non-negative.
Of course, if these co-
efficients are not non-negative integers, then there is no difference set corresponding to this set of used to construct
Od(X)'S.
e[w](X)
Furthermore, if congruence 3.21 is being
for some divisor
expect that integral polynomials
e[d](X)
have been constructed previously.
w
of
v,
it is quite reasonable to
(for all divisors
d
of
w)
will
Thus, the assumption that they are known is
not restrictive.
This assumption allows the use of the following lemma, which
helps screen the
ed(X)
candidates by imposing the condition that
e[w](X )
have
integral coefficients.
Lemma 3.9. d
of
w,
Let
w
be a positive integer and suppose that, for each divisor
an integral polynomial
integral polynomial
O[w/p](X )
6d(X )
is given.
Further, assume that an
is known, for each prime
that these are consistent with the given
O[w/p](X) ~ ed(X)
ed(X)'S,
i.e.,
p
dividing assume that
modulo fd(~)
w,
and
69
for all divisors
d
of
w/p.
Then, necessary and sufficient conditions for the
existence of an integral polynomial
e[w](X),
e[w](X) ~ ed(X)
for all divisors
d
of
w,
modulo fd(x)
Proof.
p
Let
dividing e[w](X)
(3.26)
are that a-i
%(x) ~- e[w/p](:~) for all primes
such that
w.
Here
mod(P,fwPz (x)) w = paw I
with
p
prime to
(3.~) w I.
be an integral polynomial satisfying congruence 3.26.
By the Chinese Remainder Theorem
e[w](x) ~ e~[w,/P ](x)
mod(x w/p - i)
e[~/p](X) ~ ew(x)
mod[x ~/p - l, r(x)].
e[w/p](X) -= % ( x )
mod[p, fP 1
SO
Thus a-i
(x)]
i.e., congruence 3.27 has been established. Conversely, assume
e[w](X)
Remainder Theorem from the given
is the polynomial provided by the Chinese ed(X)'S
(it may not have integral coefficients)~
then by congruence 3.24
[9[w](X ) - ew(X)] BW,w(X ) ~ 0
(mod x w - 1)
7O
(w) If
s
divides
modulo
s[e[w](X
w,
x s - i,
) .
9w(X)]
Xs - 1 x - 1
-0
(mod
xW-1)
the Chinese Remainder Theorem guarantees that
.
(3.28)
8[s](X) ~ e[w](X)
and so
w
8[w](X
)
x
x
w
s
- 1
- e r
](x)
x
Ls~
-i
x
s
-
1
(mod
xw -
1)
- i
thus (3.28) becomes
(;)s
We[w](X) ~ Wew(X) -
Let
w = P2 "'" Pj~
sEe[s
w
by
w -
~ = w/pw.
a P
Then, since
w
w
(3.29)
is glven by
a2 a. P2 "'" PjJ
~(q2r) = 0
for any prime
only non-zero terms in the sum (3.29) are those for which Hence dividing through by
(mod x w - i).
ewC~) xS-i
where the prime power decomposition of
w
and define
3(x)
s
q,
the
is divisible by
yields
"-n
T)
p~S[w](X) ~ p~Sw(X) r
r[e[r~](x)
- ew(x)]
x x
7T
w
r£o
- 1
(3.30)
-1
r~pTr
modulo
x
- i.
Thus the theorem is proved provided it can be shown that the sum
on the right side of congruence 3.30 has all its coefficients divisible by By the second part of equation (3.23), with
w I = w/pa~
pF.
71
a-I f
a-i
(x~) =~l(X p )---fPwl (x) P2"'" Pj
modulo
p.
Thus congruence 3.27 implies
%(~) - e[w/p](~) ~ o
mod
[p,f (xW)]
which in turn implies~ by congruence 3.24, that
mod(p,Z - l)
[e[w/p](X) - ew(x)] Bpv, v(~ ~) ~ 0
or
[e[w/p](X ) -@w(X)] ~
~(~)r
r'~l
As
e[r~](x) ~ e[w/p](X)
x
modulo
mod(p,x w - 1).
xW-re0 1 - 0
(3.31)
-i
x rw - i
for all divisors
r
of
~
this be-
comes
Z rlTT
Z
l
x
-1
o
mod(p,Z
l)
where it should be noted that the polynomial on the left side of this congruence has integral coefficients by assumption. Y(x),
of degree less than
~~. ~ (~) r rpp~
w,
So, the unique integral polynomial
defined by
r[e[r~] (x) - ew(x)]
xwrw- l x -1
(moaZ-
l)
72
has every coefficient divisible by
pv.
Thus
e[w](X ) = ew(~ ) _ (pv)-i
~(x)
has integral coefficients, as was to be shown. Thus, provided a complete list of candidates for each divisor
d
of
w)
ed(X)'S
is available (for
congruence 3.21 (or its computationallymore convenient
formulation, congruence 3.29) can be used together with Lemma 3.9 to construct integral polynomials
@[w](X)
and ultimately, if these have non-negative coef-
ficients, the difference set polynomial Of course, for any divisor d ~ l
e(x)
itself.
of v, the major source of polynomials
ed(X )
is the equation
ed(~)ed(~ -1) = n
which must hold for any primitive
d th root of unity
spondence between the polynomials
ed(x)
factorizations of
n
over the field of
an algebraic integer which satisfies
(3.32)
~.
Thus there is a corre-
and a restricted set of principal ideal dth roots of unity.
c~ = n
If
ed(x) ~ ~ x j ~, ai xi
(c)(~) = (n)
n
will be constructed.
ideal in the field of
(3.33)
in the field of d th roots However, for small
it can be done.
As an example of this method, the 7, 12, 14}
8d(X)'S associated
fd(x).
modulo
of unity is, in general, an extremely difficult problem. v, k, ~
is
are given by
Determining all such principal ideal factorizations of
parmmeters
i
(the bar denotes complex conjugation)
then, by a theorem of Kronecker [Theorem 2.20 above], the only with the principal ideal factorization
c = Z ai~
Here
v, k, h = 21, 5, 1 e[l](X) = k = 5
difference set
{3, 6,
and since (2) is a prime
3rd roots of unity [Theorem 2.19 above],
e3(x) = ~ 2x a
73
for some
a = 0,i, 2.
By congruence 3.27
e[l](x ) = ~ : a ~ ± 2
thus
@3(x) = + 2x a
necessarily.
mod(3, x - i)
By congruence 3.29
eE3](x) = 2x a + i~ [5- 2xa] xx 3--i1 _ ~ a + 1 + x + x 2
So, by shifting the difference set if necessary,
_ l) .
(mod x 3
813](x) = 3 + x + x 2
with the
shift fixed modulo 3. In the field of
7 th roots of unity the ideal (2) splits into a product of
two prime ideals [Theorem 2.19 above] and since, as Reuschle (1875, p. 7) lists, (i + ~7 + ~ ) (i + ~7 + ~ ) = 2, (3.32), the ideal ~73)2.
(87(~7))
these ideals are principal.
can only be (2)
Thus, by equation
62
(1 + ~7 + ~7 )
or
(1 + ~7 +
Since these last two could only correspond to equivalent difference sets,
only one of them need be considered. (b = 0,1,...,6) using
or
4
If
(@7(~7)) = (2)
then
e7(x) = + 2x b
and congruence 3.27 shows that the sign is negative.
@7(x) = -2x b
a contradiction.
However
eEv](X),
in congruence 3.29 yields negative coefficients for
Thus, without loss of generality, one may assume that
e7(x) = ± (1 + x 4 + x6) 2 x c
for some
c = O,1 ..... 6.
5 - + (i + x 4 + x6) 2 x c e + 9
By congruence 3.27
mod(7, x - i)
thus the negative sign prevails and from congruence 3.29
e[7](x) -z _xC(l + x + 2x 3 + 2x 4 + x 5 + 2x 6) + 2(i + x + ''' + x 6)
The different values of (3,7) = l, c
c
(mod x 7 - i).
correspond to different shifts of the set.
can be specified arbitrarily without affecting
uniquely specifying the shift one may assume that
(c = 5)
e[3](x);
Since thus by
74
e[7](x ) = 2 + x 3 + x 5 + x 6
In the field of two prime ideals ~i)
= 2,
be (2) or
these ideals are principal.
1>2
(I + ~ i + ~
d = 0,1,...,20.
3.27;
When
d = 3e.
~21(x)
21 st roots of unity,
~[3](x)
For
thus
P = 7
P = 3,
the ideal
.18,2
(i + ~21 + ~21 ) " @21(x)
e[21](x)
for some
e = 5.
Thus
[3, 6, 7, 12, 14} Of course,
is the desired difference
x 21 - i.
(mod
x 21 - i)
set.
there are much simpler ways to construct this particular difference
set (Lemma 3.7 for exsmple), all the principal
but the method is perfectly general and works whenever
ideal factorizations of
unity can be determined.
@d(X) = ~
(n)
in the fields of
d th roots of
Thus there are parameter sets where this method is
easier to apply than any of the others. is a square and
modulo
3.29 yields
e(x) : Q[21](x) e x 3 + x 6 + x 7 + x 12 + x 14
i.e.,
can only
3.27 rules out the minus sign and shows
this same congruence yields
by congruence
+
The first two of these do not
= (I + x 12 + x18) 2 x i5 ~ x 9 + x 15 + x 18 + 2(x 3 + x 6 + x 12)
Computing
•
(@21(~21))
= ~ (i + x 12 + x18) 2 x d
congruence
2
(i + ~1221 + ~21 )18 (i + ~ i
As before,
12
or
= 3 + x + x
the ideal (2) splits into a product of
[Theorem 2.19 above] and since
satisfy congruences
that
and
~n x s
As Turyn (1960) noted,
for all divisors
d
of
v
the cases where
n
are particularly
nice. Note i.
In the course of the exsmple above the fact that a trial
had a negative, eT(x ) = -2x b.
though integral,
@[7](x)
coefficient was used to exclude the possibility
This is a consideration
outside the range of L e n a
might be led to suspect that the complete collection of conditions
3.9.
Thus one
(3.27) imposed
by Lemma 3.9 was not sufficient to guarantee the existence of a difference This is false. the problem,
That is, given that the polynomials
i.e., that
@d(X)
set.
used are meaningful
for
75
Sd(X)Sd(X'l)
=
In k2
then any integral polynomial polynomial of a
e[v](X)
when
d % 1
(3.34) when
x = 1
computed b y the above process is the Hall
v, k~ k - difference set.
integral polynomial of degree less than
e[v](X)O[v](x-l)
x = ~d
For, whatever else it is, it is an
v
~ n + ~(1 + x + " "
which satisfies
+ x v-l)
(mod
x v - I)
SEv](1) ~ k as follows from equations (3.26) and (3.34).
Thus with
9[v](X) = Z a'xil
(a i
i n t e g e r s ) i t follows t h a t (compare constant c o e f f i c i e n t s )
2 2 2 a_o + al + ''" + av--I = k
a 0 + a I + "-- + av_ I = k
and the only solutions to these diophantine equations have conditions
a i = 0,i.
So the
(3.27) of Lemma 3.9 together with the trivially necessary conditions
(3.34) are not only necessary but also sufficient for the existence of a v~ k, h - difference set. Note 2.
Since this method and that based on Lemma 3.8 are both aimed either
at establishing the non-existence or at the construction of successive polynomials 9[w](X),
they are often combined (i.e., for a particular value of
method is easier takes precedence).
w
whichever
Generally speaking however (as seen in the
example given in section III.C.) the successful application of Lemma 3.8 requires the knowledge of a multiplier or at least a w-multiplier.
[Thus, in view of the
multiplier theorems of section Iii.A~ Lemma 3.8 is more likely to be useful when
76
v
has a relatively large divisor
w
prime to
n.]
On the other hand, multipliers
play no obvious role in the method of this section [where known they can be used to restrict the possibilities for
9d(X)] ;
thus the two approaches tend to comple-
ment each other. Note 3.
The successive construction of the
~[w](X)'S
as a means of deter-
mining whether or not a particular difference set might exist has been used almost from the beginning of the study of difference sets.
Indeed, the use of the
algebraic number theoretic implications of congruence 3.32 in this, is also quite standard. 1961). ]
[Perhaps the best documented examples are in the works of Turyn (1960, Nevertheless the explicit determination of the relations (3.19)~ (3.20),
(3.21) upon which the method is based as well as Lemma 3.9 is quite recent and is due to H. C. Rumsey, Jr.
IV.
DIFFERENCE SETS OF SPECIAL TYPE
Various groupings of difference sets have been studied more extensively than others.
These groupings usually consist of all (or all cyclic) difference sets
with a certain fixed property.
For example, those having
may be constructed by some special process.
k = 1
or those which
If the common property is of a
constructive nature, the grouping is usually called a family of difference sets. (These difference set families and their special constructions are discussed in Chapter V.)
The present chapter concerns itself with some groupings of difference
sets which have received special attention but are not of the familial type.
A.
Planar Difference Sets The incidence matrix of a non-trivial symmetric block design with
also the incidence matrix of a finite projective plane.
h = 1
is
That is, if the blocks
are called lines and the objects are called points, the incidence matrix details the structure of a system of
(i)
2
+ n + 1
each line contains exactly n + 1
(ii)
v = n
n + 1
points and
v
lines such that
points and each point is on exactly
lines
any two distinct points are contained in one and only one line;
any
two distinct lines contain one and only one point in common (iii)
there exist four points no three of which are on the same line.
This last condition serves to exclude certain trivial configurations.
[An
introduction to the study of finite projective planes is provided by Albert and Sandier (1968).
See Dembowski (1968) for a comprehensive survey. ]
An open question, which has received a great deal of attention is that of determining the values points, exist.
n
for which finite projective planes~ with
They are known to exist whenever
n
2 v = n +n+l
is a prime power and known
not to exist whenever the associated Bruck-Ryser condition (see section II.B. fo~
78
this) is not satisfied.
For all other values of
projective plane is undecided.
[In particular
n,
the existence of a finite
n = lO
is undecided;
i.e.,
it is
not known whether or not a symmetric block design exists with parameters v, k, h =lll,
ll, 1.]
Primarilybecause of the interest in this problem~ the existence question for / cyclic symmetric block designs with k = 1 (i.e., difference sets with k = l) has been pursued extensively.
These difference sets are called plana r or simple.
Planar difference sets do exist with parameters h = 1
for all prime powers
tion details].
[Singer (1938),
see section V.A. for construc-
On the other hand, not all finite projective planes correspond to
cyclic difference sets; ~ny
pJ = n
v = p2j + pj + l~ k = pJ + l~
those that do are called finite cyclic projective planes.
of the results originally developed for planar difference sets were
subsequently generalized to the case of arbitrary sections of this survey.
h
and as such appear in earlier
In order to increase the readability of this section some
of these results are repeated here;
others are merely referred to when needed.
Three areas of interest regarding these planar difference sets are discussed in this report. n = p3
p
Singer's construction process for planar difference sets with
prime, is presented in section V.A.
The question (still open) of
whether there can exist multiple inequivalent planar difference sets for prime power
n
is mentioned in section VI.A and elsewhere.
Finally,
results concerning planar difference sets for general values of restricted to prime powers) are given.
in this section, n
(i.e., not
Of course, since all known planar difference
sets are of the Singer type, these results are mainly rules which establish the non-existence of planar difference sets with certain parameter values. If
t
is a multiplier of a planar difference set, then
t
determines an
automorphism of the associated symmetric block design (as noted in section I.E. above) and hence an automorphism of the associated finite cyclic projective plane v;
thus
t
the points of
is also said to be a multiplier of the plane ~
under its cyclic automorphism is
for an arbitrary automorphism
~
v.
[If the ordering of
Po, PI~...,Pv.1,
to be a multiplier of
7,
then, in order
it is necessary that
79
there exists an integer
t
according to the rule
such that the points of the plane are permuted under Pi ~ Pti "]
All non-trivial planar difference sets have non-trivial multipliers as is easily seen from Theorem 3.1 and the parameters In particular,
all divisors
(Hall, 19h7) and
3
t
of
n
v~ k, h = n
are multipliers.
2
+ n + l, n + l, 1.
[In fact, the primes
2
(Mann, 1952) are multipliers of a planar difference set if and
only if they divide
n.
This is not true in general;
multiplier of the Singer set
v, k, k = 21, 5, 1.]
for
ll ( ~ 25 )
Since
(k,v) = 1
is a for all
planar difference sets, Theorem 3.6(2) shows that there exists at least one shift E
of the difference set (i.e., line of the plane) which is fixed by all the
multipliers.
Actually,
combining a result of Evans and Mann (1951) with one of
Mann (see Hall, 1947):
Theorem 4.1.
At least one and at most three shifts of any planar difference
set are fixed by all the multipliers. fixed shift
E;
When
n ~ 0,2
this shift contains the object
does not contain
0
when
n ~ 2
modulo
3.
0
If
modulo for
3
n ~ 0
n ~ i
modulo
there is a unique modulo 3
and there may be one or three shifts fixed by al__~lthe multipliers. determined by the pair of objects determined by the object pairs multiplier satisfies
Proof.
Since
t ~ i
Vl, 2v I
O, v I
modulo
and
n ~ O, 2
modulo
shift is unique and no divisor of the blocks e.g.,
[0])
[a, ha, n2a].
fixed shift n,
(save
E.
When
O, 2v I
v
O, 2Vl;
3
then
n
modulo
also divides
3
then
t ~ i
(n - i, v) = i
n
n - i. contain
3,
0
thus the fixed
3
elements each,
belongs to the unique so
v = 3v I.
Since
(t,v) = i
necessarily,
Here
fixes the three objects
and hence also the three shifts containing the object pairs
Vl, 2v I.
3.
This implies that all
(n - i, v) = 3;
modulo
or
yields the first part of
(n - i, v) = i;
Thus determining whether or not
as well as any other multiplier
O, v I, 2v I
The shift
are fixed if and only if every
it follows that
fixed by the multiplier
n ~ i
v = 3v I
3.
v = (n - l)(n + 2) + 3
When
and
is always fixed, whereas the shifts
Thus applying Theorems 3.4 and 3.5 to the multiplier the theorem.
then
3
O, Vl;
the only other multipliers
8O
possible have
t e 2
modulo
thus the only shift fixed by containing
Vl, 2v 1.
With
3. n
These fix
0
and interchange
vI
with
2Vl;
and also fixed b y such a multiplier is the one
Thus the theorem is proved.
n = 2,3,4,7
the planar difference sets of Singer (see section V.A.)
provide an example for each of the possibilities listed in Theorem 4.1.
Corollar~f 4.2
(Evans and Mann, 1951).
trivial planar difference set with t
modulo
while if
v
n ~ 2
Proof. case 3 E
divides one of modulo
Since
divides
v
3
n, n + i. then
So
n ~ O,
fixed by all the multipliers. v
uniform size
As
~.
~
If
If
divides
2
t
is a multiplier of a non-
+ n + i
modulo
Furthermore
prime, then the order
n ~ 0
modulo
3
n ~ i
then
modulo
~
of
divides
3
(for in this
and hence there is a unique shift (since
v
is prime), all the non-
are distributed into disjoint blocks E
3
~
n + I.
is prime it follows that
v).
zero residues of
v = n
2
[a, ta,...,t~-la}
of
is necessarily a union of these blocks with the possible
addition of the object
O
the only question is whether or not
O
belongs to
E.
So the corollary follows from Theorem 4.1. Another important result, used but not explicitly stated by Hall (1947), was stated b y Mann (1952) as
Theorem 4.3. and if
If
ti~ t2, ts, t 4
tI - t2 ~ t 3 - t4
modulo
v,
are multipliers of a planar difference set then
(t I - t2)(t I - t3) ~ O
Proof.
modulo
v.
Theorem 4.1 guarantees the existence of a shift
E
fixed b y all
multipliers.
Such a shift must contain, together with each of its elements
the elements
tle, t2e , t3e, t4e;
thus
tle - t2e ~ t3e - t4e
modulo
v.
e,
81
Since
h = I
element
e
this is only possible if of
E
tie = t2e
or
i ~ e i - ej
ated equations
modulo
(4.1) for
v
for some
e. i
and
Applying Theorem (4.3) with
modulo
el, ej
e. j
of
2
and
2j - 2 ~ 0
modulo
divisor
of
t
v.
n
thus subtracting the associ-
2j + 1
i.e.,
t I = l~ t 2 = 2, t 3 = 2 j - l, t 4 = 2 j 2
(4.1)
t I = l, t 2 = 2, t 3 = 2 j, t 4 = 2 j + 1
for which they are not distinct multipliers,
planar difference sets having
E;
v.
establishes the theorem.
the only planar difference sets having
Similarly
So for every
it follows that
(t I - t2)(t I - t3)e ~ 0
Now
tle = t3e.
and
2j - 1
shows that
as multipliers are those
2j - 1 ~ 0
modulo
v.
yields the result that the only as multipliers are those for which
When Theorem 4.3 is combined with the fact that every
is a multiplier for any planar difference set (Theorem 3.1),
it provides a quite effective non-existence test for these difference sets in the case where
n
is composite.
Many of these cases fall within the scope of the
following corollary:
Corollary 4.4.
(Evans and Mann, 1951)
Let
al, a 2, a 3
be non-negative
integers for which
aI a2 a3 q - Pl = P2 - P3 '
where
q' PI' P2' P3
are prime divisors of
no planar difference sets with
Proof.
Since
aI Pl < 3q,
n~
plq~
v = n
2q
aI
2
n
and
a2 P2 < 3q
Pl % q % P2"
Then there are
+ n + i.
it follows that the hypotheses imply
a2
n2
which contradicts Theorem 4.3 and establishes the corollary.
82
This corollary modulo
6
[let
Another
shows,
for example,
that no planar difference
set has
n ~ 0
q = 3, Pl = P2 = P3 = 2, a I = a 2 = I, a 3 = 0].
efficient
non-existence
test for planar difference
composite
n
is provided by the next theorem.
collection
of some of the more easily recognizable
Of course,
these special
full generality.
[Actually
cases were established
[See Hall (1947)~
Mann
(1952),
it is no more than a
special
before
sets having
cases of Theorem 2.9.
Theorem 2.9 was known in
Hall and Ryser
(1951) and Evans
and Mann (195m)]. Theorem 4.5. and
q
t
be a multiplier
be prime divisors
ing conditions
are met
of
n
n
and
t
has even order modulo
(ii)
p
is a quadratic
(iii)
n ~ 4
or
6
(iv)
n m i
or
2
8
modulo
4
2 nI
or
modulo
2 n I + n I + i.
The elementary
[see, for example, (i)
modulo
If the order of
(iii)
~ery If
n m 4,6
modulo
8.
is
(iv)
~
and
of
q
P m 3
(n,v) of
modulo
then
2f = i
q 8
modulo
4
and
is of even order
p
residues
t f =- -i
modulo
is used in this
q,
then
n
is even
symbol q
so Theorem 2.9
suffice.
is of even order modulo
So the Jacobi
least one prime divisor residue,
q
(1951)].
t
non-residue
p
Then if any of the follow-
number theory of quadratic
Nageli
set and let
a square:
2 nI + nI + i
modulo
and the fact that (ii)
of a planar difference
respectively.
non-residue
n ~ nI
Proof.
v
is necessarily
(i)
(v)
proof
Let
of
~ v
(let
p = 2)
= -i.
for which
q.
~us 2
and
v ~ 5,3
there exists at
is a quadratic
non-
i.e., this reduces to (ii).
the reciprocity
law for the Jacobi
symbol and the ~ p o t h e s e s
2
)=-l
dove
83
since
p
divides
divisor (v)
q
Since
n
2
nI
2 n~ + n~ + l; As an example
n
13
excludes
and
2
nI
it
+ nI + i
shifts.
case
D
if a planar difference
(Hall, v
2 vI = nI + nI + 1
1957).
designate
v.
nI
Since
+ nI
p
of even order modulo
+ 1.
is of even
some prime divisor
n I = 3.
set is to exist with
or
12
modulo
13.
This shows that n ~ 3
or
9
Among others this
an individual multiplier
Let
t
the finite
(t - 1,v) = v I t.
subplane
Vl
Of course
having
of
is a multiplier
nI + 1
be a multiplier
t
set
m a y fix more than
vI
may be
vI > 1
0,v'~...,(v I - 1)v'.
let Let
Then
E
representation
~v' = e i - ej
is unique
(i ~ ~ < Vl).
it follows
that
representations
of
~v'
unless
also fixes
these elements
of
is uniquely
v'
from
E
D
vI > 3
then
determine
a
every multiplier
h
D
in section III.B. are which is fixed by
is of course also a planar difference
v',
non-zero multiple
Further,
be a shift of
of
E
but if
for general
as well as all multiples from
3
then the fixed elements
E = [el,...,en+ I}
t.
or
shifts of
Do
Vl"
v = VlV' ,
the multiplier
1
vI
together with the fixed shifts
The first assertion was established
Assume
plane generated by
and exactly
points on a line.
of the subplane
of a planar difference
cyclic projective objects
and the fixed objects
cyclic
E;
2
modulo
n = 35.
fixed by the multiplier
above.
~ 0
In fact
and let
Proof.
of a proper prime
Theorem 4.1 shows that at most 3 shifts of any planar difference
Then there are exactly
D
v
(v) of Theorem 4.5 when
p ~ 2,4,5,6,7,8,10,11
Theorem 4.6. set
that
is also a divisor of
are fixed by all the multipliers, 3
follows
it is necessarily
consider
the possibility
Whereas
the existence
thus this case reduces to (i) and the theorem is proved.
must be a square
modulo
This insures
as in (iii).
or
2 nI + nI + 1
order modulo of
v
~ nI
So any divisor of
q
of
n.
t
represented
being multiples
of
v'
But since
set. t
e. - e. = te. - te. e.
and
as a difference also.
e..
So the fixes
E,
are distinct Hence every
of elements
So the elements
of
of E
84
divisible
by
v'
determine
parameters the form
E + iv'
fixed by
t
for some
constitute
as was to be proved° shifts fixed by the subplane Bruck v = n
2
t
Vl
j,
If
s
Since
t
with
nI + 1 of
Vl"
Vl' ~l
t
with
is of
together w i t h the shifts points on each line
D then
(Theorem 3.5) hence
Thus all assertions
[For, each line of
v I > 3)
s
s
permutes
the
is a multiplier
of
of the theorem have been established° plane
v,
with
2 v I = n I + n I + l,
with
contains
n - nI
Thus the number of points of
v
2 n = n1
then
points of
v
which
which lie on no line of
2 2 2 + n + i - (n I + n I + i) - (n - nl)(n I + n ! + i) = (n - nl)(n - nl)
n > nI 2 nI
than
on exactly
necessarily,
it follows that
there exists a point
2 nI + nI + I
lines joining n + i
be improved to
P
P
lines of
of
~
n ~n~o
which
to the points of ~
that in certain special cases
it must be that (including
n _> n 2 + n I + 2o
If
n
~i
are distinct
n ~n~
+ nlo]
divide
n,
see H. Neumann,
For any integer
fixes every object and shift that
are thus
nO + 1
Corollary
4.7
(Hall~
is a line fixed by
t
and as
Roth
P
lies
(1964) shows
that of Theorem 4.6) this inequality m a y
plane examples
nI
divides
(For an arbitrary
are known where
is a multiplier t
fixes.
whenever
If
E
then Theorem 4.6 shows that
objects
So the
nI
subplane
does not
1955).
j > 0, t j
(t j - 1, v) = v 0
~i"
In all known cases of Theorem 4.6
finite projective
.
is actually greater
lies on no line of
but the question of whether this must be so is open.
of an arbitrary
E
Vl
is another multiplier
has a proper subplane
for
is
2
if
fixed by
(1955) has shown that if a finite projective
+ n + l,
n
n,
set (non-trivial
Since each shift fixed by
the objects
among themselves
are not points of
Vl
difference
a cyclic subplane
also.
2 n ~ n I + n 1.
or
a planar
2 v I = n I + n I + l, k I = n I + l.
of
E
1947)o and if
fixed by
If v0
t
t j.
t
is, and
tj
is any shift fixed by
v 0 = n02 + n O + 1
certainly t
and
and that there
So
is a multiplier
is a divisor of
v
of a cyclic plane such that
7,
(t x - 1 , v ) =
if v0
85
for
x = j
nO + 1
but not for
points on
permuted by
t
E
v = m set
+
D1
of
D
mr
fixed b y
(Ostrom,
+ l, n = m r ,
Since
m3
These
lengths
1953).
of
2
nO + 1
divide
and there are exactly
points of
E
are thus
jo
If a planar difference
(r,3) = 1
vI = m
is also a multiplier
Proof.
t j.
with
with parameters
2 v0 = nO + nO + 1
then
in cycles whose
Coroilar~ 4.8 2r
x < j,
set
D
has parameters
then there exists a planar difference
+ m + l, n I = m.
Furthermore,
every multiplier
D 1.
is obviously
a multiplier
Theorem 4.6 once it can be established
of
D,
the result
follows from
that
(4.2)
(m 3 - i, m 2r + m r + i) = m 2 + m + 1 = v I.
From
(r,3) = I
it follows that
thus the only question (n + 2) + 3 3v I
shows that
is whether
value of
n
divides the left side of equation
some multiple
(n - i,v) = i
is the only candidate.
and this implies that
vI
or
If it were
vI ~ 0
modulo
3v I 3.
for which this happens.
3;
of
vI
thus then
Thus
Now
(m - l,v) = 1 m ~ 1
v ~ 0
So equation
does alsoo
modulo
modulo
or 3
9
(4.2), v = (n- i). 3.
So
necessarily
and there is no
(4.2) is valid and the corollary
has been established. Assume v. t
Then of
q
D
modulo
a planar difference
D
exists and let
is said to be a type I divisor of
having q
set
be
lower order modulo
~.
q
w
or
is the modulus of a proper subplane
Mann,
1951):
if there exists v.
some multiplier
Let this order
Then
where w
be a prime divisor of
than it does modulo
(t ~ - 1,v) = w ~ 0
properly
v
q
divides
v.
modulo
By Theorem 4.6 ~l"
q
2 w = nI + nI + i Thus,
and either
in particular
w = 3
(Evans and
86
Corollary modulus q
4. 9 .
If no proper factor of
of a planar difference
is also its order modulo Prime divisors
as it does for
v
q
then
n
3,
q = 3
q ~ i
set, then the order of every multiplier
t
modulo
for which every multiplier
q ~ 1
q
~ = 1
prime divisor
3.
so
q
(Ostrom, of
v,
If
~ = I
q = 3.
If
of
D
(i.e.,
1953).
n ~ i
~ = 3
then
~
of
modulo 3
If a planar difference group of
q,
type Ii~ so
t 3 - i =- 0
modulo
Corollary
of
q
modulo
v.
i.e.,
v
n q
q
of
modulo
q
modulo
q
and
divides
D
set
~(q)
i.e.,
D
has a type ii
is cyclic and its order
of the multiplier of
Thus
that
v).
D
Theorem 4.11
modulo
t I (t 3 - l) ~ 0 t3 - 1 ~ 0
1953).
q
be distinct
and if then
modulo
modulo
is isomorphic
q.
qo
Since
But
q
is of
the distinctness
set with
q - i. n ~ pJ
is sought
the search to the cases where
q
of
of
with a subgroup of the non-
Theorem 4.5 insures that no multiplier
any of the prime divisors
(Ostrom,
t!, t 2
it is cyclic and has order dividing
When this is done~
D
Then there exists a multiplier
t 2 ~ tlt 3
in restricting
group of
Hence this image group is
For let
4.8 shows that if a planar difference
even order modulo
q.
also and this contradicts
group of
is no loss of generality
not a square.
modulo
this implies
So the multiplier
zero residues
residues
it would follow that
necessarily
q
have the same image. tI ~ t2
t 2 ~ tlt 3
modulo
(tl, q) = 1
there
then
The image under reduction modulo
such that
tl, t 2.
n2 + n + i ~ 0
For if
then the multiplier
No two multipliers
multipliers
tI ~ t2
3.
and this implies that the order or
q;
modulo
is of course a subgroup of the non-zero
t3
has the same order
q - i.
Proof.
cyclic.
t
3o
Theorem 4.10
divides
is the
v
and
modulo
modulo
q
it should be noted that the only prime divisors
modulo
i.e.,
+ n + I ~ 3
by the prime
are called type II divisors.
n3 - i ~ 0
divides 2
are
divisible
v.
and
Before going further n2 + n + i
v
v.
n
is
is of
Thus
Suppose that there exists a planar difference
87 2
set modulo
v = n
necessarily
distinct primes.
for some
i, qi
multipliers
+ n + 1
where
3h
where
Theorem 4.12
h
and the
Then the order
is odd and
3h
n
S
and all the prime divisors
qi
v
cyclic group of multipliers
divides
n
(When
n ~ 0
divisor of
modulo
3.
pliers
n + 1 n ~ 1
when
n ~ 2
modulo
3,
residues size
of
v
S.
are distributed
Thus the unique
is necessarily
a union of these
Thus the theorem is proved.
3.
By Theorem 4.1 the object
Combining
a result of 0strom
Theorem 4.13. v = VlV2, v I > 1 w i t h respect to
by
t
and
S
divides
v.
v 2 > l, Suppose
then
S
Proof.
divides
Let
are multiples
E
of
n - nI
0
further that
type II divisor
and
[a, ta,...,tS-la]
is added if and only if
(t j - 1,v) = Vl, Then
addition of n ~ 0
(1951) yields
set with
v2
is of type II
where
j < S
and
2 vI = nI + nI + 1
t
and
v2
are primes,
nI ~ 1
and
tJ;
it contains
nI + 1
objects which
(see the proof of Theorem 4.6 above). E~
let
a(t ~ - i) ~ 0 of
t
vI
S
divides
(n - nl)/V 1.
objects of
q
vI
let
4.1) shift fixed by all the multi-
where every prime divisor of
be a shift fixed by
v2
Then
sets
(1953) with one of Evans and Mann
If in addition
S;
is of type II the
Suppose that there exists a planar difference and
n - nlo
v.
of the
is always a type I
v
into disjoint
is a generator of the cyclic group of multipliers.
modulo
3
S
sets together with the possible
modulo
other
then
3
group is cyclic of order
(Theorem
0.
n - nI
modulo
Since every prime divisor of
the object
divides
of Theorem 4.11 are satisfied
are of type II, then the order
By Theorem 4.10 the multiplier
be a generator of this group.
of uniform
~(qi ) = qi - 1.
v)°
Proof.
non-zero
are not
is not a square and that
divides
If the hypotheses
of
qi
of the cyclic group of
1953)o
when
(Ostrom,
... qj
Suppose further that
is of type II.
is
v = qlq2
v 2.
Thus
~
If
be the least power of
modulo ~ = S
v I v2 and
hence S
t
a
such that
t~ - 1 z 0
divides
is any of the
n - n I.
modulo If
v2
at ~ a some is
88
prime then
(Theorem 4.10)
S
divides
v~~ - 1.
2 n 2 + n + 1 = (n I + n~~ + l)v 2
Since
(n - nl)(n + n I + i) V~
-
i
=
vI
Now
vI
(since
divides
n - nI
n I ~ 1).
divides
by assumption,
Since
(n - nl)/V 1
S
divides
thus
n - nI
n + n I + 1 =- 2n I + 1 ~ 0 and
vI
is prime it follows that
and M a n n to
sets with
(1951) to
n < 100 have
n
parameters
n, v
Let
respectively.
Corollary
(b)
If
n = 4,6
modulo
8
then
(c)
If
n -= 1,2
modulo
4
and if
(d)
If
n - nI
n2
modulo
tests for
P = 3
then
(e)
If
p
is a quadratic non-residue
of
(f)
If
t
is of order
and
if
n - 0
If
t
(g)
n + 1 (h)
Let
is of order if
n -= 2
v = vlq
3. ~
v
n
v
3o
If
n
of any multiplier
If
is not a square
~
t
be prime
set with conditions:
p
n
is a square.
is of even order (In particular
the
were used.) q
then v
n
is a square.
is prime,
v
~
then
then
is not a square
modulo
must be odd.
~
~
~
n
must divide
divides
must be odd.
set exists modulo q
divides
must be odd.
is prime,
where no planar difference
~
then
is a square.
and n
4
and if
is not a square
modulo
modulo
order n
If
of a difference
modulo
n2 + n I + 1
modulo
p, q
must be a square.
n I = 1,2~3~5,6,7
~
and let
Theorem 4.3 must hold.
n
n 2 + n I + l,
modulo
p. 209) was pushed
is subject to the following
4.4 and more generally
with respect to
(1968,
be a multiplier
Then the existence
(a)
or
that the only planar
This was extended by Evans
to Dembowski
t
v , k , h = n 2 + n + l, n + l, 1
associated
S
(unpublished).
The tests of Evans and Mann. of
a prime power.
n < 1600 and according
n _< 3600 by V. H. Keiser
divisors
v1
as was to be shown.
Utilizing mainly Theorem 4.3 Hall (1947) established difference
modulo
v 1.
Then the
G = (q°(Vl), q0(q)).
89
(i)
Let
v = vlq ,
Vl, q
exists modulo
vI
multiplier
t
of the
modulo
but
of
t
vI
modulo
divides
Proof.
are both primes and a planar difference set
but not modulo
t ~ 1 q
q.
Let there exists a non-trivial
v,n + 1,1 - difference set such that modulo
divides
n - n I (n I ~ l)
(n - nl)/V I
above.
where
q.
n - nI
If
n
is not a square then the order
and is odd.
then the order of
t
If, in addition, modulo
(f)
Consider test
and
(h).
(g)
v
thus
~
divides
So Theorem 4.5 finishes off.
t
~(Vl)
and
Consider test (i).
are prime it follows from Corollary 4. 9 that multiplier
divides
(a),o..,(e)
are a combination of Corollary 4.2 and Theorem 4.5.
Here Theorem 4.6 shows that the order of
the same as its order modulo definition.
q
v1
and is odd.
Theorem 4.3, Corollary 4.4 and Theorem 4.5 establish
Tests
t = 1
q
t
modulo
~
divides
Since
q
is a type II divisor.
of the test corresponds to the multiplier
tj
vI
is
~q) and
by
v1
The
of Theorem 4.13.
Thus Theorems 4.5 and 4.13 establish the validity of test (i). Since the only known planar difference sets have prime power
n
and in fact
may all be constructed by the process of Singer (see Chapter V), it is easy to make up perfectly reasonable conjectures about planar difference sets.
Just conjecture
that any property possessed by the Singer sets holds for all planar difference sets.
One such property of the Singer set
v,k,h = p2j + pj + l, pJ + l, 1
that its multiplier group consists of all the powers of powers of
p
p
modulo
are necessarily multipliers has been observed earlier.
only the powers of
p
are multipliers
v.
is
[That all The fact that
is non-trivial aud is due to Gordon~ Mills
and Welch (1962), see section V.A. for a proof of their more general result. ] a general planar difference set (i.e.,
n
For
not necessarily a prime power) the
analogous conjecture would be that its multiplier group was generated by the prime divisors of
n.
A reduction lemma established by Halberstam and Laxton (1964), in
the course of providing an alternate proof
(h = 1
only)
of the Gordon, Mills~
Welch result on Singer sets, may be of some use in attacking this problem. that
t, nt, n2t
Note
are necessarily either all multipliers or all non-multipliers of
9O
any planar difference set with be of reduced type modulo
n
2
+ n + 1
0 < r, s < n
Lemma 4.14. with
n > i,
and
modulo
0 < r + s < n.
j
to
n2 + n + i
Then
(Halberstam and Laxton)
then at least one of
define an integer
if
j-=r+sn
with
Furthermore,
k = n + i.
If
t > i
t, nt, n2t
and if
(t,n 2 + n + i) = i -
is of reduced type modulo
2 n
B.
+ n
+ i.
Hadamard Difference Sets Difference sets whose parameters
Hadamard difference sets.
(ii)
There are several reasons for this; among them are:
the relative abundance of such difference sets with
k < v/2
as usual,
section I.B.
h
varies between
i
and
(v - 3)/4~
see
Thus planar difference sets and Hadamard difference sets
present the extreme values of (iii)
are called
Like the planar difference sets of section IV.A., these
have been extensively studied. (i)
v, k, h = 4t - l, 2t - l, t - 1
h.
the autocorrelation function of the
l, -1
characteristic function of
these difference sets (see section I.D. for this) is minimal [~(j)
= -1
for
j ~ 0
modulo
communications applications
v].
This has led to several digital
Ksee, for example, Golomb e t a l
(1964)
or Goldstein (1964)]. (iv)
the relation between these difference sets and the~ as yet unsolved, Hadamard matrix problem (see the note below).
The known Hadamard difference sets can be classified according to the value of
v. (a)
The groupings are: v = 2 j - i, j ~ 2;
section V.A. discusses a large family of difference
sets whose parameters include these (construction details are given there).
91
(b)
v = 4t - i
is prime;
here there always exists a Hadamard difference
set composed of the quadratic residues modulo prime
v = 4t - 1
is expressible
as
v
4x 2 + 27,
set (due to Hall, 1956) whose construction
(see section V.B.) and when the there is an additional
is discussed
difference
in section V.C.
Some
others exist also. (c)
v = p(p + 2)
where
p
and
p + 2
are both prime numbers
(see section
V.D. for the details). Occasionally
v
satisfies more than one of these conditions
most part, this leads to multiple (a) Mersenne
and
(b)
(a)
overlap if and only if
primes of the form
Chowla and Lewis and
4x 2 + 27
difference
v
are
sets.
Specifically: the only
is a Mersenne prime;
v = 31, 127
and
[see Skolem,
131071
(1959) for this].
(c)
overlap if and only if
The known Mersenne ~rimes with
inequivalent
and, for the
v = 15.
(i.e., primes of the form
2 j - i)
are
v = 2j - 1
j = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281,
3217, 4253, 4423, 9689, 9941, 11213 at present,
[see Gillies,
1964;
in fact
2 I1213 - i
is,
the largest number known to be prime].
The difference v = 3, 7, 15
and
sets corresponding 31.
Now
(a) provides a difference set corresponding
to
v's
set and (b) provides is equivalent
sets arise from
While all known Hadamard difference
two more.
(c).
In particular,
sets with parameters
except for
is prime, thus
However,
the difference
to the one from (a);
thus only two
v = 31. sets have
not all of them can be constructed by the methods in (a), (b),
are inequivalent
v = 31 = 25 - 1 = 4.8 - i = 4.12 + 27
4x 2 + 27
truly distinct difference
to these
v's of types
(a),
(c),
(section V.A. - V.D.) indicated
there are exactly six inequivalent
v, k, h = 127, 63, 31
(b),
difference
(see Baumert and Fredricksen,
1967),
three of which do not arise from these constructions. It is known (Golomb, v < i000,
then
v
Thoene, Baumert)
is one of the forms
that if a Hadamard difference
set has
(a), (b), (c) with six possible exceptions.
92
These exceptions are decide
this
v = 399, 495, 627, 651, 783
and
975.
The methods used to
were~ of course, those of Chapter II and III.
Some problems of electrical network theory [Belevitch (1968), see also Goethals and Seidel (1967)] have led to the consideration, skew Hadamard difference sets.
That is, Hadamard difference sets which contain
precisely one of the residues
d, v - d
for
1 < d < v - 1.
obvious examples of such sets are the quadratic residues primes
q ~ 3 (mod 4).
Theorem 4.15.
e(x)
x
v
Let
Johnsen (1966B) has shown that there are no others.
(Johnsen)
The only cyclic difference sets which are skew
T(x) = 1 + x + ''' + x v'l
for such a difference set.
- i
and so [using
Of course, the
(see section V.B.) of
Hadamard are given by the quadratic residues of a prime
Proof.
among other things, of
q,
where
q ~ 3 (mod 4).
and consider the Hall polynomial
It satisfies
e(x)e(x "l) ~ n + h T(x)
1 + e(x) + e(x -1) ~ T(x) and
@(x)T(x) ~ k T(x)]
modulo it
follows that
e2(x) + e ( x ) + n ~ n T(x)
So, for all
v th roots of unity
mod(x v - 1).
~ % i
e2(~)
+ e(~)
e(~) = ( - i ±
since
v = 4n - 1 Since
for which Let
p % q
q
divides
v
p
roots of unity.
,.,~)/2
(4.4)
it has an odd prime divisor but
q
s+l
does not divide
be another odd prime divisor of
equation (4.4) shows that th
(4.3)
in any Hadamard difference set.
v ~ 3 (mod 4), s
+ n = o
(-1 ~ ' ~ ) / 2
Contradiction.
v
and let
q v
with and
s
q m 3 (mod 4) is an odd integer.
~ = ~p = e 2wi/p,
then
is an algebraic integer of the field of So
v
is an odd power of the prime
q,
where
93
q ~ 3 (mod 4). Let
~q = e 2vi/q
and let
z(x) =
so that
Z(~q) =
1 ~
~q
in section III.D.
divisors
d
v (=qS).
(4.5) it follows that
G i(x) = q
Here
gi = ~ 1
From (3.19),
+x
+x
~
To do this we need the complete set of Equation
(4.4) provides us with these,
el(X ) = k = (qS . 1)/2
and that
1 ~
)
(3.20) it follows that modulo
v
ew(x) Bv,w(X)
~(i) = l, ~(q) = -i and
for from (4.4),
for
-v
and
-
r = w/q.
in assuming that
2v
ew(X)
s1 = + I.
x---f:-i--!
~
rl w
~(qi) = 0
i = 1 ..... s.
for
r
r
r
x
i = 2,...,s,
So interchanging
- 1
this last summation
the order of summation and
matching terms yields
+
for all
xv - 1
wl v wll
r = w
ed'S
process
i-1 + el q(S-l)/2 z(xq
+
only involves
(4.5)
~ge shall use the constructive
and there is no loss of generality
=
Since
x[(q-l)/2] 2
+ "'" +
(Gaussian sum).
discussed
of
l
~
1
v
- - -
1 v
Oq(X)
qi i=l
1 --i xq - 1
x---?'i
{eqi(X) - eqi+l(X)} ]
.
94
Now, for
i = 0,1,...,s,
x q i ( x-V- i ' "i )
xV l -i
xq - 1
m°d(xV - i)
xq - 1
Using this fact
e(x) -= ev(X) + ~
v
-d
x v - i~ (4.6)
s-1 +
-
v
) -
aj+l(~/2)}
x q3 - 1
so the only question is whether this has integer coefficients. if
v = q,
then (4.6) becomes
e(x) ~ - 1
~
j =!
(gl = i,
+z(x) = - ~ 1
+ ~z
If
s = l~
i.e.,
as noted earlier)
+x
+x 4
+ ...
+ ~[(q-Z)/2]
2
which does indeed have integer coefficients and in fact is the quadratic residue difference set for Let
s
efficients,
q ~ 3 (mod 4).
be odd and
s > 3
then in order for (4.6) to have integral co-
it is necessary that the terms
_q(S+l)/2 2~
( ~ Z -- i I ) and
-i
q(s+l)/2 ( 6 1 z ( x ) } ( x v_- _1 )
v
(4.7)
x q- i
compensate for each other, since they are the on]~v terms of (4.6) which have the coefficient
~(s+l)/2 v
i q(s-1)/2 "
95
A l l other terms of (4.6) contain higher powers side of (4.7) contributes contributes
to
to all
(qS + qS-1)/2
other and therefore
q
S
of
q.
coefficients
coefficients.
no such difference
But the term on the left
whereas
the other term of (4.7)
So they cannot compensate
set exists.
for each
Thus Theorem 4.15 has been
established.
Note:
A n y symmetric block design
parameters course,
v,k,h = 4t - i, 2t - i, t - i
gives rise to the name Hadamard
set of a cyclic design. a Hadamard distinct matrix,
design,
-i.
a matrix of O.
orthogonal. achieves
If
-i
not necessarily
difference
That is, a
21
Such a matrix
abs. val.
det.
= (4t)2t].
the matrices, For example, modified
v,k,h = 7, 3, i
Hadamard
difference
of any two
+l's
to this
row inner products 4t
are
whose rows are mutually
a Hadamard matrix because determinantal
Thus the name Hadamard
of
in the incidence matrix of
whose distinct
to the associated block designs the
0
row and column of
the upper bound specified by Hadamard's
which,
set for the associated
square matrix of order
is called
design,
is such that the inner product
Thus, by adding a constant +l's is constructed
cyclic) with
is called a Hadamard
is used instead of
the resultant matrix
rows is
uniformly
its determinant inequality
[i.e.,
passed from the inequality
and finally to the difference
difference
set
(1,2,4]
to sets.
has associated
incidence matrix and Hadamard matrix:
-i -i -i 1 -i 1 1
i -i -1 -1 1 -1 1
i i -i -1 -1 1 -i
Chapter 14 of Hall existence Spence
(i.e.,
-i i 1 -1 -i -1 1
i -i 1 1 -1 -i -i
(1967)
-i i -i 1 1 -1 -i
surveys
of Hadamard matrices;
(1967),
and Whiteman
Goethals
(1970).
i i i 1 1 1 1 1
-i -i 1 -1 1 1 -1
i -i -i -1 1 -i 1 1
(with proofs) for results
and Seidel
i i -i -i -i 1 -i 1
i i i -i -i -i 1 -i
i -i 1 1 -1 -i -i 1
i i -i 1 1 -i -1 -i
subsequent
(1967), Wallis
cases are
n = 188,
i -i -i 1 -i 1 1 -i
most of the known results
on the
to Hall's book see
(1969 AB,
1970),
It is known that the order of such a matrix
and the first few undecided
i -i i -i 1 1 -i -i
Turyn is
236 , 260, 268, 292.
(1970)
i, 2
or
4t
96
C.
Barker Sequences ~ Circulant In 1953 R. H. Barker
munications,
considered
and minus ones coefficients
Hadamard Matrices
(1953),
in connection with a problem in digital
the question
[bi] [ ~
of the existence
of finite
with the property that their aperiodic
should be as small as possible,
sequences
comof ones
auto-correlation
That is, he asked that
v-j c.J =
~
b i bi+j
= 0
or
-i
i=l
for all become
J, i ~ j ~ v - i. customary
sequences~
and
to relax Barker's
whose aperiodic
Barker sequences. denotes
He found such sequences condition
v = 3, 7, ii.
It has
slightly and call all finite
autocorrelations
Only the following
for
e. J
are restricted
Barker sequences
to
are known:
I, -i
-i~ 0r ix (+
denotes
+i
-i)
v =2
++
v=4
+++-;
V = 5
+ + +-
+
V =7
+++-
- +-
v =ii
+++-
° - +-
v =13
+++++-
++-
+
- +-
- ++-
+-
+
together w i t h the sequences which may be derived from them by the following
trans-
formations:
b: = ~"C-l) i b 1 1 b: = (-i) i+1 b. l 1 b~ = -b.. l I
In fact,
Storer and Turyn
(1961) have shown that any further Barker sequences which
may exist must be of even length,
indeed they show that
v ~ 0 (mod 4)
is necessary.
97 Note that in terms of the
i, -i
representation of the characteristic function
of a difference set (see section I.D. for this) all these sequences correspond to difference
sets.
v = 7,11,13
For
v = 2,3,4,5
the difference sets are trivial;
the sets have parameters
v,k,h = 7, 4, 2;
ii, 5, 2
and
for 13, 9, 6
respectively. It can be shown (Storer and Turyn, 1961) that Barker sequence.
Further,
cj + cv-j = ~ ( j )
if
v - O (mod 4)
then
e. + c
,]
v-j
~ v (mod 4)
cj + Cv_ j = O.
in any
Thus, as
[the correlation coefficient defined in section I.D], any
further Barker sequence
{bi}
has autocorrelation function
Rb(J)= Ii if j ~0
modulo
v
otherwise
i.e., a two-level autocorrelation function. difference set. 2.1 and since v = 4N 2. values of
Since n, v
Since ~(j)
v
is even,
n = k - k
= v - 4(k - h) = 0
k(k - l) = h(v - l) show that
Thus such a sequence corresponds to a
for
is a square (say j ~ O
modulo
v
N 2)
it follows that
for any difference set and
k = 2N 2 - N
or
2N 2 + N.
b y Theorem
n = k - ~
Since these
these
k's correspond
to complementary difference sets, there is no loss of generality in assuming that v,k,h = 4N 2, 2N 2 - N, N 2 - N. Thus, further Barker sequences exist if and only if there exist difference sets with parameters (v,n) = ( 4 ~ , ~ )
v,k,h = 4N 2, 2 ~
= N2 ~ i
- N, ~
- N
for
and 2.17 above rule out many of these cases; with the single exception
by a constructive method;
Since
here, this is a subcase of the unsolved problem con-
cerning the existence of cyclic difference sets with
i ~ N ~ 55
N ~ i.
N = 39.
(v,n) > i.
in particular,
N ~ 55,
all the cases
Turyn (1968) excludes this case
essentially that of section III.D.
Barker sequences exist they must have
Theorems 2.13
i.e.,
Thus, if any further
v ~ 12,100.
A matrix is called circulatory or said to be a circulant if each successive row is derived from the previous row by shifting it cyclically one position to the
98
right.
For example, the matrix
is a circulant.
(+
for
.
÷
+
+
+i, -
÷
for
-i)
÷
+
This particular circulant has every entry
orthogonal to each other.
!1
and its rows are
Matrices with these properties are called Hadamard
matrices and they have been extensively studied [see the note at the end of section IV.B].
The example, thus, is a circulant Hadamard matrix.
rearrangement and scalar multiplication by only known circulant Hadamard matrix.
-i
In fact, up to
of its rows and columns~ it is the
It follows immediately from the autocor-
relation function of a Barker sequence of even length
v ~ 4,
that there is a
one-to-one correspondence between such sequences and circulant Hadamard matrices. Thus (from the Barker sequence results above), if there exists any further circulant Hadamard matrices they have orders
v ~ 12,100.
It sho~Id come as no surprise
then, that the absence of any further Barker sequences/circu!ant Hadamard matrices has been conjectured.
[Using the fact that
-i
is never a multiplier of a non-
trivial cyclic difference set (Theorem 3.3 above), Brualdi (1965) has shown that there does not exist an Hadamard matrix of order
v ~ 4
which is a symmetric
circulant.] The related problem of finding
i, -i
sequences of length
v
for which the
maxi~2~m aperiodic correlation coefficient is of least magnitude (i.e., for which
m~ lojl J is minimized) and indeed the problem of determining this minimum, at least asymptotically as a function of
v,
is unsolved.
of the known results on this subject.
Turyn (1968) provides a survey
V.
FAMILIES OF DIFFERENCE SETS
The known difference sets (with a few exceptions) can be divided into families.
This chapter deals with these families of difference sets, construction
methods specific to them, their multiplier groups and the status of some open questions related to them.
A.
Sin~er Sets and Their Generalizations.
The Results ' of Gordon~ Mills and Welch
Singer (1938) discovered a large class of difference sets related to finite projective geometries.
These
parameters:
have
N+I v
=
~
N -i
q - i
for
N > i
k =
q
'
N-I ~ ~
~ =
q
q - i'
and they exist whenever
q
-1
(5.1)
q - i
is a prime power.
In order to discuss Singer's result properly it is necessary to know some of the theory of finite fields [see, for example, van der Waerden (1949) for proofs]. For any prime power
q
there exists a finite field with exactly
q
elements.
This field is unique up to isomorphism and is called the Galois field of [written
GF(q)].
The multiplicative group of
generated by any of its
~(q- i)
GF(q)
elements of order
is cyclic; q - i.
elements are called primitive roots and if
~
ever
the residues
u
is prime to
q - i.
For prime
p,
structed from
GF(q)
irreducible over all divisors
j
by adjoining any root
GF(q). of
m.
The subfields of
GF(p). ~
These generating
GF(pm),
O,i,...~p - I p;
p
~u
when-
form a
this field is often
m GF(r)~ r = q ~
of any
elements
thus it is
is a primitive root so is
field with respect to addition and multiplication modulo taken to be the generic representation of
q
can be con-
m th degree polynomial prime, are
GF(p j)
for
f(x)
lO0
GF(qm) GF(q).
is often represented by the set of all m-tuples with entries from
In this representation addition is performed componentwise but multiplica-
tion is more complicated. polynomial
am_ 1 x
m-i
Associate with the
+ ... + alx + a 0.
m-tuple
am_l, am_2,...,al, a 0
the
Then, in order to multiply two m-tuples,
multiply instead their associated polynomials and reduce the result modulo any fixed
m th degree polynomial
f(x)
irreducible over
GF(q).
The coefficients of
the resulting polynomial constitute the m-tuple which is the product of the original two.
For multiplicative purposes it is more convenient to represent
terms of a primitive root
~;
in which case,
GF(qm)
GF(qTM) in
consists of
m
O,l,C~, 2 ~ o . o ,(~q -2
m
Multiplication then becomes a simple matter of reducing exponents modulo
q
but addition is more complicated.
Both these representations of
are used
in the proof of Singer's theorem.
Table 5.1 shows both kinds of representation for
TABLE 5.1.
0
0000
O0
1
0000
Ol
o( 2 o~
J c~
6
J 8
2 lO
c~
~5
GF(26)
with
GF(qm)
f(x) = x 6 + x + i.
i01000
~i
i001
Ol
010011
~2
0010
0 i
48
001101
~49
011010
i00111
0000
i0
i00110
o~ 3
0100
io
0001
O0
001111
~4
i001
00
~0
ii0100
ooio
i i
~51
i01011
OlOi
i 0
~2
010101
~53
i01010 010111
0010
O0
011110
a35
0100
O0
iiii00
~36
i000
O0
0000
ii
22
iii011
~Y
lOll
0 0
ii0101
~38
OliO
1 l
J4
iioi
i o
~5
i01110
0001
i0
i01001
~39
0011
O0
010001
0
iOli
i I
J6
011111
0110
O0
i00010
~41
Olii
o i
~57
iiiii0
000111
J2
iii0
i o
~8
iiiiii
001110
~43 44
iioi
i i
iiii01
i011 OllO
o i 0 1
~59 6o (~ 61
ii00
i 0
~62
i00001
ii00
00
i000
ii
0001
Ol
25 26
28 c~
011100
0010
i0
iii000
J5
0101
00
ii0011
(~46
iii001 ii0001
- i
i01
GF(26).
[In this example the primitive root
no means necessary;
~
satisfies
That is, a primitive root of
will always satisfy an irreducible polynomial of degree polynomial is said to be primitive of degree find such polynomials.
polynomial of degree
m
m
such that
over
GF(p).]
The finite projective geometry
that in
dimension
and
Thus any set of
a subspace of 0
PG(N,q)
PG(N,q),
PG(N,q)
GF(q).
of dimension GF(q)
(baN, baN.l,..o,ba0) j + 1
GF(q);
such a
The trick is to
of dimension
J.
The
N - 1
Thus there are and
over
b ~ 0, b
(N + 1)-tuples determines
(qN+l _ 1)/(q - i)
subspaces of
( J + l _ 1)/(q - l)
sub-
Any two distinct hyperplanes
so they have
v = (qN+l _ 1)/(q - l)
k = (qN _ 1)/(q - i)
GF(q), con-
are identified for all
are called hyperplanes. N - 2;
N
subject to the restriction
linearly independent
intersect in a subspace of dimension points in common.
over
a primitive irreducible
are the point ~ of this geometry and the
spaces of dimension
planes in
over
pm < 109,
(N + 1)-tuples of elements of
(aN,aN_l,...,a0) GF(q).
m
m
GF(qm)
Alanen and Knuth (1964, Table 7) list, for each prime
and all exponents
sists of all
This is by
however it is often quite convenient and it is theoretically
always possible to arrange things this way.
p < 50
f(x) = 0.
(qN-1 _ 1)/(q - l) points and
v
hyper-
points in any hyperplane.
Singer
(1928) has shown:
Theorem 5.1. PG(N,q)
Considering the points as objects and the hyperplanes as blocks,
forms a symmetric block design with parameters given by (5.1) above.
block design is cyclic;
thus the points of any hyperplane determine a
This
v, k, h
difference set.
Proof.
The discussion above shows that
A,
the incidence matrix of this
configuration, satisfies
AAT=
(k - ~)I + kT,
~
= kJ.
Thus conditions (ii), (iii) of the block design definition follow from the result of Ryser (1950) discussed in section II.C. just below Theorem 2.3.
So
PG(N,q)
102
forms a symmetric block design as indicated. It remains to be shown that there exists a numbering of the points and hyperplanes of
PG(N~q)
which demonstrates the cyclic nature of the design.
be a primitive root of
G F ( J +l)
and let
~
satisfy the irreducible polynomial
f(x) = x N + I + c ~ x N + ... + C l X + c o
over
GF(q).
GF(q).
Then each power of
Since
vi
belongs to
~
(5.2)
corresponds to a unique
GF(q)
for all
i,
the elements of
are
point of
Thus there is a one-to-one correspondence between the elements
= 0,1,...,v - l)
PG(N,q)
an exponent
and the points of
~j+vi
GF(q)
and it follows that
~(j
and
(N + l)-tuple over
0,1,~v~...,~ (q-2)v PG(N,q).
~
Let
PG(N~q)
correspond to the same
which assigns to every point of
j, 0 < j < v - 1.
Consider the mapping
~:~
i
~
i+l
~:0~0
(5.3)
or in additive notation, using equation (5.2),
: (aN,...,al, a 0) ~ (aN~ I - aNCN,.-o,a 0 - aNc !, - aNCo) •
(5.~)
This mapping obviously maps points onto points and [as is clear from (5.4)] maps subspaces onto subspaces without any loss of dimension. planes into hyperplaneso that corresponding to If H
~
H,
~J~
the mapping
is not cyclic of order
and an integer
of
Since the point corresponding to
then
ts.
chosen;
so
s, 1 ~ s ~ v - l,
So
t
~j+iv
is cyclic of order
such that
~s
fixes
form an orbit in
H
H.
divides
v
on the points.
Let
under
i
~s
be a point with
is necessarily the least positive integer such that
This is, of course, independent of which element t
is the same as
on the hyperplanes then there exists a hyperplane
i, i+s ,o..,~ i+ts = ~ i
i( ts _ !) = 0. divides
v
~
Thus~ it maps hyper-
k.
Since
v
does not divide
s
i
of
H
v
was
(by assumption) the fact
lO3
that
v
divides
that
(v,k) > l;
necessarily and
ts
implies that
and as
this contradicts the fact that ~
is cyclic of order
Singer has been established. (say
(v,t) > i
~i,~,...,Gm),
v
t
divides
v - qk = 1.
k
Thus
on the hyperplanes.
it follows v
divides
s
So the theorem of
Thus, if one lists the elements of any hyperplane
their exponents form a difference set with parameters given
by (5.1) above. Any mapping
L
from
G F ( J +I)
onto
GF(q)
which satisfies
L(b~ + c%) = bL(~) + cL(v)
for all 6, ~ from
in
G F ( J +l)
GF(q N+I) to
GF(q).
such a linear functional in
PG(N,q).
and all
b, c
in
The set of elements L
[i.e., such that
Further, every hyperplane of
functional from
GF(q)
G F ( J +l)
to
GF(q).
~
is called a linear functional of
GF(q N+l)
L($) = 0]
PG(N,q)
N = 5
then
v, k, ~
6th degree polynomial (say
are
constitutes a hyperplane
is annihilated by some linear
Thus to apply Singer's Theorem one merely
computes the null space of a single linear functional. and
annihilated by
63, 31, 15
x 6 + x + 1 = 0)
and
~
over
For example, when
is to be a root of a primitive GF(2).
Consider the linear
functional
L
which maps each element into its right-most component.
shows that
L
annihilates
i
when
i
q = 2
is one of
Table 5.1
l, 2, 3, 4, 5, 7, 8, 9, 10, 13,
14, 15, 17, 19, 20, 25, 27, 28, 29, 33, 34, 36, 37, 39, 42, 46, 49, 50, 53, 55, 57. Thus these numbers constitute a difference set with parameters
Note:
v,k, 7~ = 63,31,15.
Singer's construction can be varied in several inessential ways, the
only effect of this is to generate a difference set equivalent to the original one. For example, if a different hyperplane (or linear functional) is used this merely shifts the set. G,
If the primitive root a t, tr - 1 mod(q N+l- 1),
the equivalent set
root is needed. is not in
GF(q)
rD = {rdl,.. o,rdk)
results.
In fact not even a primitive
It is only necessary to have an element for
i = 1,...,v - 1.
[The use of
difference set as that generated by the primitive root
is used instead of
~
6( = ~u)
for which
Gi
results in the same Gu+jv,
the existence of
104
which (for some value of
j)
is guaranteed by the condition on
G. ]
Extensive use will be made of the following well-known fact:
Lemma 5.2. GF(q m)
Let
L
be any linear functional, not identically zero, from
to its subfield
linear functional from defined by
GF(q j) GF(q m)
L~(~) = L ( ~ )
and let to
for all
~
be any element of
GF(q j) ~
is of the form
in
GF(qm).
GF(qm).
Then every
L~,
where
L~
is
Moreover if
~ ~ v
then
L~ ~ L w. [Linear algebra provides the fact that there are precisely functionals from
GF(q m)
to
GF(q j)
qm
linear
and the above process constructs
qm
distinct
ones. ] Complementary to any Singer set
D
is a difference set
D*
with parameters
N+l v =
for
N > i
and
q
q
- i q-l'
k = qN
h '
a prime power.
Here
qN-l(q_l)
(5.5)
=
j = O,l,..o,v- 1
belongs to
D*
if and
m
only if
L(~)
~ 0
root of
GF(qN+I)o
where
L
is a fixed linear functional and
Call this difference set
of generality in assuming that
L(1) = l,
D(L,~).
~
is a primitive
Note that there is no loss
for this may be arranged by taking a
different linear functional° The difference set of
D(L,~)
corresponding to the example given above consists
O, 6, ii, 12, 16, 18, 21, 22, 23, 24, 26, 30, 31, 32, 35, 38, 40, 41, 43, 44,
45, 47, 48, 51, 52, 54, 56, 58, 59, 60, 61, 62. that modulo 3,4,5~6,7,8)o
9
they constitute In fact
D(L,~)
4
Examination of these residues shows
copies of the trivial difference set
gives rise to the array of Table 5.2
E = [0,2,
i05
TABLE 5.2.
where the
(i,j)
that, for each
entry is i
J
0 1 2 3 4 5 6
i~= 0
i 0 1 0 0 1 1
1
D(L,G)
0 0 0 0 0 0 0 0 1 0 0 ! I i
3 4
0 1 1 1 0 1 0
5 6
0 0 1 1 1 0 1
7 8
0 i 0 0 1 1 !
0 0 1 1 1 0 1
i 0 1 0 0 1 1
0 0 1 1 1 0 1
in the difference set
one of its shifts. of
1 2
if and only if
characteristic function of the
D(L,~).
A Representation of
E,
i + 9J
belongs to
D(L,G).
Note
the rows of this array are the
w,~,~ = 7,4,2
difference set
F = [0,2, 5,6]
or
Gordon, Mills and Welch (1962) have shown that the structure
always depends upon difference sets
E
and
F
in this manner.
Specifically, they prove:
Theorem 5.3. N > 1.
Let
L
Let
q
be a power of the prime
p
and let
N + !
be a linear functional from the finite field
be an integer,
G F ( J +l)
to the sub-
w
field
GF(q),
such that
intermediate field
L(1) = lo
GF(qm),
where
functional which assigns to each GF(qm), Set
m ~
which satisfies the relation
polynomials of y = x ~.
GF(q N+l)
be the restriction of
divides in
D(L,~)
and
N + 1.
GF(q N+l)
Let
~
L
to an
be the linear
the unique element
L0(~(~)5 ) = L(~5)
~nd
and let
~ = ~.
D(L0,B)
respectively.
Let
for all
~ = vl~. @(x)
and
Let ~(y)
[For m = i,
~(~)
5
in ~
in
GF(qm). be a
be the Hall take
~(y) = i.]
Then
G(x) ~ ~(x) ~(y)
where
L0
v = (qN+l - l)l(q- i), w = (qm _ 1)l(q- l)
primitive root of
Let
Let
(mod
x v - l)
(5.6)
lO6
~(x)
: Z
xiy ri
(5.7)
and this summation is taken over those values of
~(i)
~ 0 ,
In the example above E
0 _~ i < { ,
r. 1
and
~
for which
~(czi) : B
N + 1,m,q,p = 6,3,2,2,
is the difference set determined by
The
i
-r. 1
(5,8)
F = D(L0,B) = {0,2,5,6]
and the extension
GF(qN+I)
determine the shifts associated with the various copies of
Proof. a~L(~8),
Consider 8
in
~.
Let
GF(qm),
~
be an element of
GF(qN+I).
is a linear functional from
Lemma 5.2, there is a unique element, call it
~(~),
of
GF(qm).
in the array.
Then the mapping
GF(qm) of
F
while
to
GF(qm),
GF(q).
So, by
such that
=o(Z(O~) = ~ ( ~ )
for all
5
in
GF(qm).
[L0
is not identically zero since
is a properly defined mapping from remains is whether or not of
GF(qm)~
a
and
b
~
fixed~ and let
= L((aq
:
~
to
~, ~
belongs to
L0(1 ) = 1 GF(q)
when
GF(qm).
that m : l,
L0
L(1) = I°]
+ b~)8)
Let
= L(aqS)
Thus
The only question which a, b~ 5 be elements
be fixed elements of
GF(q N+l)
then
+ L(b~8)
Lo((a~(q ) + bZ(~))8) .
is indeed a linear functional and in fact
follows from
Let
GF(q N+l)
is a linear functional.
Lo(~(a q + b~)8)
So
(5.9)
~(i) = i.
is the identity mapping on it follows from (5.9) that
When
m = i,
GF(q); ~ = L
it
since
in this case.
lO7
w-~
5jyj
~(y) =
{ 0 if L0(~J) = 0 with
j=O
5. = J
I if Lo(~J) ~ 0
and let
v-Iz 8(x) = i=0
" a.x ~ l
{ 0 if L(Gi) = 0 with
e.1 =
~-i = ~ xi~i(y)when
i if L(Gi) { 0
w-i o]i(y)= ~, si+~jy3o
i:o
j:o
Since ~ is a primitive root of GF(qTM)
every value of ~_LL(i) is either 0 or
-r.
a power of ~, say ~ I. Now si+__~j= 0 if and only if L(G i+~j) = 0 and
I
0
L(~i+~j) = L(~i~j) = L0(~((~i)~j) =
if
~(~) : o
if
Z ( i ) ~ o.
J_ri ) L0(~
So
o
if
Z(~ i) : o
3-r.1 Thus ~i(y) = 0 if ~(i) = O; while if ~(Gi) # 0
w-i ~i(Y) =
w-i
w-i
Z £i+~J yj = ~ 5J-r. yj = ~ 5J yj+ri = y 1 j=O j=O j=O •
~(y)
(mod
Xv
- l).
r.
Hence @(x) -~(y) 7. xly m the theorem is proved.
r. 1
where the sum is taken as prescribed by (5.8), and
108
Now
~(y)
is the Hall polynomial of the
has parameter values (for
w :
@0(y )
set
D(L0,~ ) which
m > !)
m q - 1 q-1 '
m-i ~ = q ,
Gordon, Mills and Welch show that, if polynomial
w~,~-difference
$(y)
~ : qm
-2
(q-l)
(5.10)
is replaced in (5.6) by the Hall
of an arbitrary difference set having parameter values (5.10),
i.eo, if
eo(X ) ~ ~(x) ~o(y)
then (5.5).
e0(x )
(mod
is again the Hall polynomial of a
x v - i)
(5.11)
v, k, h-difference set with parameters
For let
w-1 ~o(Y) =
be a Hall polynomial and let
y = x ~.
Z j:O
Let
~JYJ
G(x)
be given by (5.7).
k
e.
x
eo(X) = n(x) ,o(y) :
Then
i
i=l
where the modulo
x
e. I v
are distinct modulo
v
by construction.
Further~ by definition,
l,
Co(y) ~o(y -I) =- (~ - ~) + ~(i + y + ... + yW-l) _ ~(y) @(y-l) °
Hence, modulo
x v - i,
Io9
eo(X) eo(x'l) = n(x) ~(x-1) ~o(y) ~o(y-1) ~ ~(x) o(x"l) ~(y) ~(y-1) = o(x) e(~-1) .
Thus
el, ...,e k
example, let
[For
form a difference set with the parameter values of (5.5).
@0(y ) = 1 + y + y2 + y5
then using
G(x) = i + x2y 6 + x3y 3 + x4y 4 + x5y 4 + x 6 + x7y 6 + x 8 4
from set
D(L,~)
above, in congruence 5.11 yields the
D(L,~)
This difference set is not equivalent
as is explained below.]
Gordon, Mills and Welch show that two (5.11), are equivalent if and only if the
difference
0, 2, 6, 7, 9, ii, 12, 15, 16, 18, 22, 23, 24, 26, 30, 38, 39, 40, 41, 43, 44,
45, 48, 49, 50, 51, 53, 56, 58, 59, 61, 62. to
v,k, k = 63,32,16
w,~,~-difference
v,k,h-difference
@0(y ) = yS @(y).
sets are shifts of each other.
sets derived from
That is, if and only if
[Since
-1
is never a
multiplier of a cyclic difference set (section 3.1) the difference sets are never shifts of each other. generate inequivalent
Thus the polynomials
v,k,h-difference
the inequivalence of the two
@(y)
and
sets by these means.
63, 32, 16
@(y-l)
D
and always
This accounts for
difference sets mentioned above.]
Several lemmas are required for the proof of this result (which is Theorem 5.12 below). Following Gordon, Mills and Welch, let C = {Cl, C2,...,c ~}
be two
w,~,u-difference
B = {bl, b2,...,b ~} sets, and let
ci
,
i
be their Hall polynomials.
[If
m = i
and
i
let
~b(y ) = ~c(y ) = i.]
Put
-D
ii0
eb(~) = ~(x) %(y),
Then
eb(X )
say
B
and
such that
and
8c(X )
C.
If
are the Hall polynomials of two
B
(t~v) = 1
ec(X) = a(x) ~c(y).
and
~
are equivalent then there exist integers
Proof.
(mod
B
(mod x v - l )
~b(y ) ~ yS 9 c ( t )
(mod
and
C
such that
(5.13)
B
r.
and
x a 8c(X ) = Z xa+tiy
Let
j
be such an
i;
and
~.
a + th m j
modulo
~.
i
tr. i ~c(yt)
for which
~(i) ~ o,
then, comparing terms in (5.12)
i a+th t rh Wc(yt) x iy r. ~b(y ) ~ x y
Since
@b(y ) ~ yS @c(yt)
Now
s
yW _ l) °
are inequivalent then so are
where these summations are taken over those values of
where
and
By construction
•
where
and
(5.12)
r
O(x) ~ x r O(xt )
eb(X ) = Z xly I ~b(y )
0 < i < ~.
x v - i).
If (5.12) holds, then there exists integers
In particular if
a
and
eb(X ) ~ xaec(Xt )
Lemma 5.4.
v,k,h-difference sets,
(mod
x v - 1 = yW _ 1
(mod
x v - l)
this yields
(5.14)
yw _ i)
s = tr h - r. + ~-l(a + th - j). 3 @b(y) @b(y -I) ~ ~ - ~ + ~(i + y + ... + y w - l )
modulo
yW . i
and this
iii
implies that
~b(y )
is relatively prime to
yW _ i.
So from (5.12), i.e., from
~(x) ~b(y ) ~ xa~(x t) ~e(y t)
(mod
xv -
e(~)
(mod
xv
i)
and (5.14)
~
x a -ys
9(x t)
-
1)
and the lemma is proved. Let
Q = GF(q)
Lema
5.5.
and let
Q*
be the set of all non-zero elements of
Suppose (5.13) holds and
be an element of
GF(qN+I).
Then
~(w)
(t,v) = i.
Let
belongs to
Q*
~ = ~r,
Q.
and let
if and only if
w
~(h t)
does also.
Proof.
Since
~
is a linear functional from
G F ( J +l)
to
GF(q
TM)
it follows
that
~(x) = ~. xiyri = ~
x i+~ri =
~
xj
(5.15)
j in S
where
S
j
is the set of all
such that
~(aJ) = i,
O ~ j < qN _ 1.
is a primitive root of
Q*,
the effect of adding
by a primitive root of
Q*.
Thus (5.15) can be written as
V~
a(x) ~
~,
v
to
(mod
xj
j
Since
is to multiply
c~v ~(~)
x v - i)
j in S'
where (5.13)
S'
is the set of all
Z(~)
belongs to
len~na is trivial.
If
Q*
~ ~ 0
j
such that
~(eJ)
if and only if
belongs to
~(r+jt)
Q*,
does also.
the result follows by putting
~ = ~.
0 < j < v. If
~ = 0
the
112
Lemma 5.6. element of
Suppose (5.13) holds and
GF(qm)
if and only if
and let
~(q t)
~
belong to
belongs to
GF(qN+l).
Let
~ ~ 0.
belongs to
Q*.
By Lemma 5.5 this is true if and only if
Here
~(~w t)
~(~)
let
~ be an
belongs to
~Q*
~tQ..
Let
which is equivalent to
~ = r,
Then
Proof.
Q*~
Then
(t,r) = i.
~(~)
belongs to
L(~w t)
~Q*
if and only if
being a member of
belongs to
vtQ *
for some
v
in
first part of this proof
~(~)
is an element of
~(~-I)
~(~ t~-t)
belongs to
~tQ.. Next suppose
GF(qm).
If
v ~ 0~
~(~) = 0.
then by the
vQ*~ contradiction.
So
v = 0
and the proof is complete.
Lemma 5.7. Suppose that (5.13) holds and elements of
GF(q N+I)
ei, ai( 1 < i < s)
(t,v) = i.
Let
which are linearly independent over
be elements of
GF(qm)
s
~1,~2, .... ~s
GF(qm)°
be
Let
such that
t
s
(}.16) i=l
Then
a.
1
Proof.
belongs to
c~ Q*,
l
Since the
linear functionals
~i
K. J
i=l
i < i < s°
are linearly independent over
over
GF(qm)
such that
i
By Le~ma 5.2 these exists elements
u. J
of
K.
if
GF(qm)
( l ~ i~ j ~ s)
i = j.
can all be expressed in terms of
GF(q N+I)
there exist
such that
Z(uj q)= ~ij"
~,
that is there
i13
Then~ by Lemma 5.6j Now
~L(~ujt ~ ) = 0
~(uj Z e i ~i ) = cj,
if
i ~ j,
and
~(~u t ~ )
belongs to
Q*.
so that
~(~u~( ZC i ~i )t)
belongs to
c~ Q*.
On the other hand, using 5.16,
i
and thus belongs to
Len~na 5°8. for
c I
=
c 2
. . . . . t
al~ a 2
t % ai ~i = 0
with
ai
shows
that
=
0~
this
in
are linearly independent over
N + 1 >mo
Then
=
a 2
Let
as was to be proved.
~i' ~2' "'" Is
be a basis
are also such a basis.
GF(qm)o a I
Suppose (5.13) holds with
c~ Q*
(t,v) = 1.
then
and suppose of
is an element of
GF(qm);
c s
Lermma 5. 9 . G F ( J +l)
aj
~t,~t,...,~ ts
over
Let
~1~2 =~v,f:,...,~ s
So
Suppose (5.13) holds with
G F ( J +I)
Proof.
ajQ*.
Apply Lemma 5°7 with
. . . . .
GF(q m)
a s
=
0o
Hence
and form a basis for
(t,v) = i.
Let
w
OF( +I)
be an element of
(1 + ~)t = al + a2 t
for some elements
Q*.
Proof.
First assume that
a basis for
G F ( J +l)
over
~
is not an element of
GF(q m)
C 1 = C 2 = l, e 3 = o.o = c s = 0;
so
which contains i + ~ = Z c i ~i"
GF(qm).
Then there exists
~l = l, ~2 = ~° Moreover
Let
~'~''°"~s t
is
also a basis (Lemma 5.8) so
(I + ~)t = Z ai~ ~
with to
ai
in
GF(qm).
By Lemma 5°7,
a3 = a4 . . . . .
a s = 0,
and
al, a 2
belong
Q~.
NOW suppose
~
is in
GF(qm).
Let
~ be an element of
GF(q N+l)
such that
114
~({) = ~ and so
and
does not belong to
m Now
for some elements
Z(~)=
~(~) = ~. Q*o
~(~ it ) Thus
M
~(1) = 1
[Such a
{
exists since
does not belong to
b, e
+ {)t)
(1) ~(~(1 + {)t)
(2)
Q*.
belongs to (1 + w) t
Q*
~(q(l + {)t)
(1 + ~)t =
Thus
: ~(~(h + c{t))
belongs to
and
guarantee the existence of such an
and so
of
~(1) = 1
GF(q m)
On the other hand, by the first part of this proof
~(~(i
and
and
where
~(1 + ~) = 1 + w
by Lemma 5.6.
b + c{ t
{ = w + M
GF(qm)o
while
= b ~(~) + c ~(~ {t)
~(G ~t)
is of the form
belongs to a I + a2 t
tQ.
since
with
al; a 2
in
Combining this with (i) above completes the proof. Lemma 5.9 is the first major step in the Gordon, Mi!is~ Welch proof of Theorem
5o12 below.
To complete the proof some results about the implications of the
condition
(l + ~)t = al + a2 t
are required. every pair such that G F ( J +I)
If (5o17) holds for all w
PI' 02
of elements of
bl, b2,...,b u
(P I + 02 + ... +
GF(J+I),
GF(qN+I),
(PI + P2 )t = b! P~ + b2 P~" there exist
in
then it follows that for
there exist elements
By induction, given any
in
Ou)t
(5.17)
Q*
bl, b 2
pl, P2,.o.,pu
of
Q*, in
such that
t + t t = blP I b2o 2 + ..° + buO u o
(5.18)
Write (5.17) in the form
(I + ~)t = rw(l + s
with to
rw, s w Q
in
Q*.
Since
then neither does
w
t
(t,v) = i, and hence
t)
it follows that if rw, s
w
does not belong
are uniquely determined°
ll5
Lemma 5.10. elements
Let
N ~ 2
al, a 2
of
Q*
(i)
if
wt
T t ~ ~t
(ii)
if
G
and, for every
in
GF(J+i),
such that (5.17) holds.
is a primitive root of
For uniquely determined
let there exist
Then
are linearly independent over G F ( J +l)
which are not elements of Q, s
Proof.
w
Q~
and if
= 1 and
then s
s~/w = s~/T s /w.
= l,
then for all
w
(1 + w) t = 1 + wto
bl, b2, b3, Cl, c2, c3
of
Q*
blG~t + b2 Tt + b3~t = (~ + T + ~)t = Cl(W + T)t + c2~t
= clwt(1 + T/w) t + c2 ~t = c3(wt + sT/w Tt) + c2 ~t .
So
s iw = b2/b I
and by symmetry
S~l ~ = b3/b2, S~l w = b31b I
which establishes
part (i) of the len~ao Consider the
sw = i
assertion of (ii).
is not contained in any proper subfield of over Put
Qo
Hence
w = u,
i, t
and
I ~ u ~ J+l
2t
. I
Since
(t,v) = I
GF(J+I).
Hence
it follows that t
has degree
are linearly independent over and induct on
s 2 = s
s
Uo
Q
(as
G
t
N + i
N + i ~ 3).
By part (i) it follows that
= i
G
so
or
2.
Let
positive integers less than
uo
Since
as
sw = i
when
(t,v) = i.
i, t ,
ut
u = i
So
i
and
wt
or the elements
u ~ 3
and suppose that
sW = i
w
is not an element of
Q
are linearly independent over
i, G 2t, Gut ,
Let
i, ~ t ,
Gut
be independent
(j = i
and the induction hypothesis
S
=S
=S
neither is
s.=l
or
i, t , 2),
t w ,
Now the elements
are linearly independent over
if both these element sets are dependent then so is the set tradiction).
Q.
for all
G2t
Q;
for
(con-
then by part (i)
116
and so the first assertion of (ii) has been established. Since over
Qo
N + 1 > 3,
there is a
Then~ for suitable
~
such that
Cl, c2, c 3
in
l~ t
~t
are linearly independent
Q*.
(1 + ~ + {)t = Cl + c2 t + c3{t
with
Cl, c2, c3
uniquely determined.
by the linear independence of Qo
So
(i + ~)/~
is not in
i, t , Qo
Since ~t
s
= i, (i + w) t = rw(l + t ) ,
it follows that
a = r(l+~)/~.
and from this that
Theorem 5.11.
Hence r e = 1.
Let
v = (qN+l _ l)/(q - i) that for every such that
Proof. where
G
that and in
~
+ ar t
let
and let
be an integer relatively prime to
t
q
be a power of the prime
there exist non-zero elements Then
t
Without loss of generality
0 < t < v
and
is a fixed primitive root of
GF(qN+l).
Since
(t',v) = lo
Furthermore~ for any
w
p,
let Vo
al, a 2
Put in
p
GF(q)
modulo
v.
(i + G) t = r (i + sGGt), s~
t' = t + vc.
G F ( J +l)
Suppose
in
is congruent to a power of
e, 0 ~ c < q - 1.
cI = c3
as was to be shown.
N + i > 3,
G F ( J +l)
for some
+ a~t
By the same process it follows that
(1 + ~)t = 1 + t
(1 + ~)t = al + a2 t.
sG = v c
Q*
in
c I = c2° So
is not in
Thus
(i + w + ~)t = a(1 + ~)t + a~t = a r
where
(i + ~)t/~t
and
is in
Q*
it follows
Then 0 < t ' < J + l - 1
there exist
r' W
and
s' W
such that
(l+~)t'=r~(1+s'~t')o Note that
s' = io
Thus by part (ii) of Lemma 5o10
(i + w) t' = I + w
t !
(5o19)
117
for all
w
power of
of po
which are not elements of
q
N+l
- q
(5.19) by
roots°
Therefore
~u(1 + ~)u
yields
t' > q
= (l+~)~u
for all
~
of
G F ( J +I)
q
N+l
< 3q - 2
power of
p
the proof.
and
N + 1
N + I~ ~(x)
N+l
Let
t' +
u >
(Gordon, Mills and Welch)
~(y),
there corresponds a If
B
and
equivalent if and only if
Proof.
with
Start with respectivelY.
(t,v) = 1
Hence, since
B
B
and
Hence
(qN+l
2u k J + l
t' - l,
p
_ qo
So
- q)
N + 1 ~ 3, q ~ 2o
If
modulo
Thus
v,
t'
is a
which completes
q
be a power of a prime
N + 1 >m
~ 23
where
C
are
B
C
let
m
set
v,k,h-difference
set B
w,~,~-difference
sets then
B
and let with Hall
with Hall polynomial B
and
C
are
Co
and construct the Hall polynomials
is equivalent to
and
divides
~ = v/w,
w,~,~-difference
p
~
Ob(X), Oc(X )
then there exist integers
a,
such that
N + i > 3,
C
Let
is a cyclic shift of
and
is congruent to a power of B
~
To any
Ob(x) =- xao c(x t)
of
is not a
N+l u = q - 1 - t t.
Let
be given by (5°5) and (5.10),
be the polYnomial given by (5.7)°
B, C
- qo
Qo
is congruent to a power of
v, k, ~, w, ~, ~
O(x) = fl(x) @(X~)o
t
- 1 =
be a positive integer such that
polynomial
of
t'
Finally, the result promised above can be established.
Theorem 5.12. let
Suppose
+ (l+~)u
which is impossible since t
N+l
which do not belong to
q
or
Q.
Then (5o19) becomes a polynomial equation of degree at most
with at least Multiplying
GF(q N+I)~
(mod
x v - l) o
Lemma 5.4, Lemma 5.9 and Theorem 5.!1 establish that p
modulo
vo
Now every power of
p
t
is a multiplier
(applY Hall's Theorem 3.1 above to their complements).
So
ll8
*c(y t) -yU*c(y ) for some integer
Uo
Thus by Len~na 5°4
*'b(y) ~ y
and
B
(rood yw_ i)
is a cyclic shift of
¥c~y )
C
(mod
yw _ i)
as promised.
In the course of the above proof it was concluded that, for
Gb(X ) ~ xaGc(Xt )
only happens when
t
(mod
is congruent to a power of
same observation shows that only powers of sets the
~o
With
D(L,~)
powers of p
m = 1
p
p
x v - l)
modulo
v.
If
B = C
p
may be multipliers of
As mentioned earlier, Theorem 3ol shows that the
are always multipliers of these difference sets~ so:
Theorem 5.13.
(Gordon, Mills and Welch)
difference set or if
D
If
D
is a non-trivial Singer
is any non-trivial difference set derived from congruence
5°6, then the multipliers of
D
are precisely the powers of
p
modulo
Another by-product of Theorem 5o12 is the existence of parameters for which at least and let
J
j > 0
inequivalent difference sets exist°
denote the number of inequivalent
given by (5o10)o
Since
-1
2J
which is a cyclic shift of any other. v, k, Z-difference sets with
Theorem 5o14o
of
v, k, h
possible
v, k,
For, let sets with
w,~,~-difference
Hence there are at least given by (5.5) or (5.1)o
(Gordon, Mills, and Welch)
be positive integers with r
w,~,~-difference
vo
m _> 3~ w, ~,
is never a multiplier of a non-trivial difference set
(see Theorem 3~3) there are at least
m, M
this
may be multipliers of the difference
this shows that only the powers of
difference sets.
N+I>3,
m > 3.
Let
Let
q
inequivalent
In particular
be any prime power and let
N + 1 = ~u
prime numbers, not necessarily distinct.
2J
sets, none of
and let
M
be the product
Then there exist at least
2r
ll9
inequivalent
difference
sets with parameters
(5.5) and thus
(by taking
complements)
with parameters
N+I v_q
B.
Nthpower
Residue
A difference or of the
-1 q-l'
Difference
set which
is some reason for distinguishing
restricted
to divisors
residues
and the
that the
f
of non-zero venient
5.15.
form a difference
Proof. residue
(N, v - l )
coincide°
rl, oo.,r f
for the number theory required difference
When
is a prime,
set with parameters
residue
srl, sr2, sr
r,
say
[Nagell
residue
r I - r 2 =- ro
If
residues
Thus one of s
also.
equally often as a difference
and vice versa.
r 2 - r I - -r)
residue with the same number of representations v , k = 4t - l, 2t - l;
The set
~1,2~4~
modulo
7
thus
The best known (1933):
residues modulo
v
is a quadratic r I - r 2, r 2 - r 1 residue of
Thus eve~# equation
v
set w i t h
the fact
(1951) is a con-
is any quadratic
residue of
(e.g.,
N th power
sets of Paley
c, -c
st. - s r . -= sr l j
these congruences
sets,
v, k, h~ n = 4t - l, 2t - l, t - l, t.
with an equation
Reversing
set.
form a subgroup of the group
corresponds
is represented
the
v = N f + l,
the quadratic
1 < c < v - lo
are quadratic
v,
Attention m a y be
in this section].
sets are the quadratic
v = 4t - 1
some prime
difference
= d >l
For primes
w i l l be used extensively
for a prime of this form,
then
the two types°]
Recall from number theory that exactly one of
is a quadratic v,
v
N th power residue
since if
N th power residues
residues modulo
N th power residue
Theorem
d th power residues
distinct
reference
v - l,
N th powers modulo
N TM power residue difference
is called an
when there
of
1 q-l"
Sets and C~clotomy
zero are called modified
N
N-I h=q
is composed of all the
N th powers and zero,
[Those containing
N q-1 q-l'
k
yields
also.
r. -r. -=r
Hence every quadratic of quadratic
residues.
every quadratic
non-
Hence this is a difference
h = t - lo
is the first non-trivial
quadratic
residue
120
difference set.
Note that the parameters imply that every quadratic residue set
is a Hadamard difference set (see section IV.B. for a discussion of this special type). Another class of residue difference sets was discovered by Chowla (1944):
Theorem 5o16o
The biquadratic residues of primes
a difference set with parameters
v = 4x 2 + l, x
v,k,% = 4x 2 + l, x 2, (x2 - 1)/4.
The first non-trivial biquadratic residue difference set is for v,k,~ = 37,9,2 Other
and
odd, form
x = 3,
here
D = {1,7,9,10,12,16,26,33,34}.
N th power residue difference sets have been discovered and a general
theory for them has been developed by ~mma Lehmer (1953)o
In order to explain her
results a little of the theory of cyclotomywill have to be introduced.
No more
cyclotomy than is necessary for an understanding of the difference set results of sections V.B., V.Co, and V.Do is developed here.
[A complete introduction to
cyclotomy from the classical point of view is given ab initio in Dickson (1935 ABC); T. Storer's booklet "Cyclotomy and Difference Sets" (1967 A) gives a different development also ab initio and discusses most of the difference set results of these sections.] Let
v = Nf + 1
An integer
R
number R
f
g
be a fixed primitive root of
is said to belong to the index class
exists an integer sists of
be an odd prime and let
x
such that
distinct numbers
(~,m)N
R ~ gNX+~ (mod v).
~.
That is,
R + 1
(~,m)N
with respect to
g
if there
Thus, the index class
g~,g N+~, .o. ,gN(f-1)+~
counts the number of times
belongs to index class
~
modulo
v.
Vo
~
con-
The cyclotomic
belongs to index class is the number of solutions
m
when x, y
of the congruence
g
where the integers
Nx+~
x, y
that there are at most
+ 1 ~
gNy+m
(mod
are chosen from N2
numbers depend not only on
0,1,...,f- 1.
v)
This congruence shows
distinct cyclotomic numbers of order v, N, ~, m
(5°20)
but also on which of the
N
and that these ~(v- l)
121
primitive roots
g
of
v
is chosen.
The following elementary cyclotomic facts are all that is needed for an understanding of the results of this section (note that when
f
is odd,
N
is
necessarily even):
(~,m)N = (~',m')N
when
~ ~ ~'
and
m ~ m'
(mod
(m,~)N
N)
f
even
f
odd
(5.22)
(~,m)N = (N - ~, m - ~)N = I
(m + N/2,
l
N-I Z
(~,m)N = f - n~,
where
m=0
~ + N/2) N
II
~ ~ 0 (mod N)
f
even
L
~ ~ N/2 (mod N)
f
odd
n~ = < 1 0
then
(~,m)~ s
is prime to
Proof. nition.
is based on the primitive root
(5°23)
otherwise
(5.24)
(~,m)~ = (s~,sm)N
where
(5.21)
g' ~ g
s
modulo
v;
necessarily
v - lo
Equation (5.21), of course, is an immediate consequence of the defi-
The first part of (5.22) follows from the definition after congruence
5.20 is multiplied through by the inverse of its first term
[i.e., by
gN(f-x)-~].
Similarly, the second part of (5.22) follows after congruence 5°20 is multiplied through by
-1,
that is by
-i -= g(Nf)/2 = ~
gN(f/2)
f
even
(5.25)
L gN(f-1)/2
+N/2
f
o~d .
The sum in equation (5.23) is simply the number of successors of members of index class
~ which belong to any index class at all.
Since
-1
is the only element
122
whose successor does not belong to an index class, equation correctness of (5.23).
Equation
to
sx
v - 1
implies the
(5.25) implies the
(5.24) follows from the definition,
ranges with
x
since
s
prime
over a complete set of residues modulo
fo In terms of these cyclotomic n~nbers
(~,m)N
and sufficient conditions for the existence of
it is possible to give necessary N th power residue difference sets;
in fact:
Theorem 5.17o
(Lehmer,
1953)
N th power residues of a prime even,
f
Necessary and sufficient conditions,
v = Nf + 1
form a difference set, are that
N
is
is odd and that
(~,o) N :
(f - I ) / N
for
The parameters of such difference sets are
~ ~ o , I .... , ½ N -
v,k, ~ = v,f,
(5.26)
l o
(f- I)/No
Necessary and
sufficient conditions, that the
N th power residues and zero for a prime
form a difference set, are that
N
be even,
1 + (0,0)N = (~,0)N = (f + I)/N
The parameters of such difference sets are [Note that equations
f
for
~ = 1,2 .... , ½ N - 1 .
(~,0)N
N th power residues
then there are exactly
h
solutions
r i - rj ~ 7
and hence the congruence
(5.27)
v,k,Z = v, f + i, (f + I)/No
(5.22) and (5°24) show that these existence conditions
individual cyclotomic ntnnbers
If the
v = Nf + 1
odd and that
(5°26) and (5.27) are independent of the primitive root
Proof.
that the
g,
even though the
are not°]
rl,.oo,r f ri, rj
are a difference set modulo
v3
to the congruence
(mod
v)
(5.28)
123
riril
-= Trj I + i
for all
7 ~ 0
(c,0)N = ~; as
thus
v.
But for
(i,0)N = ~
h = (f - 1)/N
is an f
modulo
for
follows from
(mod
T
in index class
0 < i < N - 1
v,k = v,f.
If
N th power residue, hence a multiplier,
is odd and as
and
f
all
i.
Nf
is necessarily even,
is odd, equation Hence
That is, the v,k,~ = v,f, When
0
(5°22) provides
v
c
(5.29)
this implies that
and in particular f
so is
(5.26) holds
is even~ then by (5.25),
which contradicts N.
(f - I)/N
form a difference
-1
Theorem 3.3~
Conversely,
since
0)N : (f-
(i,0)N = (i + 2 '
(5°29) and (5.28) have exactly N th powers modulo
v)
N
Thus is even
I)/N
solutions for all
for
7.
set with parameters
(f - I)/N. is added to the set of
N th power residues the only effect is that
differences
r. - 0 = r. 1
and
l
have to be counted alsoo
As before,
So,
is even) and
each
f
is odd (hence ri
(N/2, 0)N
and
-r i
In particular,
v,k = v, f + lo
odd and that equation (5.22)] it follows, difference
f
l
even implies that -1
(5.27) holds.
equation
Since
that the
v,k = v, f + i;
(5.27)holds,
[1,2,4},
(ii)
v = 4x 2 + i, x
IN = 4, Chowla odd;
e.go, the
and (f+l)/N
is even, here
f
is
[by equation
and zero form a
N,
i.eo, those for which
with the following results:
sets exist which consist of (i)
the quadratic residues of primes
~=
thus
h = (f + I)/N.
Theorem 5.17 has been applied for several values of
Difference
N
N (i + ~ , 0)N = (i~0)N
hence
N/2;
(0,0)N
since
N th power residues
the eyciotomic numbers have been computed,
is a multiplier°
belongs to index class
On the other hand, assume that
as before,
set with
Theorem 5.18.
-i
is represented once more than the numbers
indicate.
follows from
N
0 - r. = -r.
l
v = 4t - i;
e.g., the
(1944)] the biquadratic v,k,Z = 39,9,2
set
IN = 2, Paley
v,k,h = 7,3,1
(1933) ]
set
residues of primes [1,7,9,10,12,16,26,33,34],
124
(iii)
[N = 4,
Lehmer
(1953) attributes
residues and zero for primes [0,1,3,9),
(iv)
[N =8,
v = 4x 2 + 9, x
Lehmer
1 = 64b 2 + 9, k = a 2, ~ = b 2 [1,2,4,8,16,32,37,55,64), (v)
IN = 8,
Lehmer
these to M. Hall, Jr. ] the biquadratic odd;
e.g., the
(1953) ] the octic residues of primes
with
a, b
odd;
e.g., the
the next such prime
v
is
v,k,h = 26041,3256,407
odd,
b
(by these methods)
even;
v < 34,352,398,777.
it can be shown that n__oomodified quadratic
sets exist and further that no non-trivial
residue difference
sets exist when
(1966), of
v.
(x)
(viii)
N = 16,
The case where
2
(vl)
N = 6,
N = 12, Whiteman
Whiteman
Lehmer
is not an octic residue is open;
(xi)
(1967),
Baumert and Fredricksen
(submitted for publication) The case where
5
for the case when
is not a biquadratic
The calculations
(ix)
(1957) for the case when
set here then (xii) 5
residue or modified
(1953),
(1960A),
that if the residues form a difference N = 18,
v = 8a 2 +
e.g., the
residue difference
(1960 B),
set
140,411,704,393 ,
set, there are no more such primes
On the other hand
set
v = 8a 2 +
v,k,h = 73,9,1
(1953)] the octic residues and zero for primes
49 = 64b 2 + 441, k = a 2 + 7, ~ = b 2 + 7, a
Whiteman
v,k,?~ = 13,4,1
2
(vii)
N = 10,
N = 14, Muskat is an oetic residue however,
it is known
v > 1,336,337. N = 20,
Muskat and Whlteman
is a biquadratic
residue of
v.
residue remains unsolved.
involved in determining
the cyclotomic numbers of these
orders are far too extensive to present here, the numbers themselves may be found for
N = 2 - 6
in Dickson
papers cited above for By way of example,
N = 8
in Lehmer
though~
if
thus
N = 2
and
(0,0)2 = (1,1)2 = (1,0)2 = ( f -
v = 2f + l, f
quadratic residue difference
f
odd, equation
1)/2
and
(5.22)yields
In the case
is more difficult;
(0,1)2 = (f + 1)/2.
So
set [ i.e., an alternate proof
and (5.27) prohibits
sets.
which may be
hence the quadratic residues of
odd, form a difference
of Theorem 5.15 has been given],
the cyclotomic numbers
and in the
(5.23) provides two linear relations,
(5.26) of Theorem 5.17 provides no restrictions, every prime
(1955 B),
N = 10,12,14,16~ 18, 20 o
(0,0)2 = (l~l)2 = (1,0)2 solved for
(1935 A),
the existence
N = 4, f
however,
they are
of modified
odd, the computation of
125
(0,0) 4 = (2,2) 4
:
(2,0)4
= (v - 7 + 2y)/16
(O,l) 4 :
:
(3,2)4
= (v + 1 + 2y - 8x)/16
(1,2) 4 = (0,3) 4 = (3,1)4
= (v + 1 + 2y + 8x)/16
(1,3) 4
(0,2)4
and the rest are
(v - 3 - 2y)/16,
where
Thus (5.26) of Theorem 5.17 requires
= (v + l
- 6y)/16
v = y2 + 4x 2
and
y ~ 1
v - 7 + 2y = v - 3 - 2y = v - 5
[That is, a proof of Theorem 5.16 has been given.]
or
f = 2j + l, x
Y = -3,
then
thus
v : 9 + 4x2.
v = 8j + 5 : 9 + 4x2.
Since
Thus
or
4. y = 1.
The existence of modified
biquadratic residue difference sets requires, by (5.27), that 2y = v + 3
modulo
v + 2y + 9 = v - 3 -
v = 4f + l, f
odd, let
8j - 4x 2 = 4, 2j - x 2 = l;
hence
is odd, which establishes part (iii) of Theorem 5.18. A completely analoguous theory of cyclotomy can be developed for prime powers
v = p (for
i
= Nf + 1
[see, for example, Hall (1965) or Storer (1967 A)].
N = 2,4,6,8)
in the same conditions for
residue difference sets as before. are not cyclic and, in fact, for
However N = 4,8
Theorem 5.18 are only satisfied for
N th power and modified
(for
i > l)
residues plus for
00
N th power
these difference sets
the quadratic conditions imposed by
i = l,
i.eo,
v
is prime.
quadratic residues, this was established by A. Uo Lebesque rediscovered by Hall (1965).
This results
[For the bi-
(1850) and recently
Hall's paper also contains a proof for the biquadratic
Storer (1967 A, Theorem 20) states this result without proof
N : 8.] The multiplier group for
N th power residue difference sets has been de-
termined:
Theorem 5.19.
(Lehmer, 1953)
multipliers of a non-trivial
Proof.
Clearly the
another multiplier,
then
The N th power residues themselves are the only
N th power residue difference set.
N th power residues are multipliers. tD ~ D + s
modulo
v,
where
t
Suppose
t
was
belongs to an index
126
class
j,
with
j ~ 0 (mod N)o
trl,.o.,trf;
so
0
D, r
is not in
s ~ 0 u
modulo
+ s ~ tr
r s u
has exactly
f
In fact, the index class
-1
solutions,
y
v.
(mod v)
+ i ~ tr s y
If
(5.26) when 0
D
belongs to
belong to index class
-1
(mod
(i,0)N = 0,
i.
Then, if
v)
(N - i,j - i)N = (i,J)N = f.
which contradicts the existence
is non-trivialo D,
Thus, by (5.25),
then s
tD ~ D + s
Do
tr u
-
-1
is a multiplier also, contradiction.
for some
consists of
implies
to
s ~ 0
s
i.eo, the cyelotomic number
But this implies, b y (5.23), that condition
Let
j
u;
(with
belongs to index class thus
t
and
-1
s ~ 0) N/2.
implies
0 - s
belongs
On the other hand,
belong to the same index class°
As mentioned in sections I.Eo and III.Ao above, every divisor of is a multiplier for all known cyclic difference sets.
Hence
n = k - h
This is not readily apparent
for the residue difference sets of Theorem 5.18, but was established by ~mna Lehmer (1955 A).
First consider the quadratic residue difference sets for prime
for these it is necessary to use the law of quadratic reciprocity Nagell
(1951)]o
whether
(= +l)
As usual, the Legendre symbol or not
Theorem 5.20.
(=-l)
p
(p/q) = ± 1
All divisors of
v,
are quadratic residues.
(mod 8),
q.
are multipliers
v = 4t - lo
By Theorem 5.19, it is sufficient to show that all divisors of
quadratic residues modulo n
will be used to indicate
n (= k - h)
for the quadratic residue difference sets for primes
Proof.
[see, for example,
is a quadratic residue of the odd prime
(Lehmer, 1955 A)
v = 4t - l;
but then
2
an odd prime divisor of
Now
which, of course, n = (v + 1)/4
(and thus of
are
follows if the prime divisors of and
2
divides
n
only if
is a quadratic residue by the reciprocity lawo n
n
v + 1 = qm)o
Let
v ~ -1 q
be
By quadratic reciprocity
127
i.e.,
q
is a quadratic residue of
v.
More generally there is:
Theorem 5.21. cyclic
C.
Nthpower
(Lehmer, 1955 A)
All divisors of
n
are multipliers for the
residue difference sets of Theorem 5.18.
More Cyclotomic Difference Sets Denote each index class
i
by
Ci,
then if
v = 31
and
N = 6
these
classes are:
Letting
CO + C I + C 3
CO + C I + C3
CO:
i,
2,
4,
8,
16
CI:
3,
6,
12,
17,
24
C2:
5,
9,
10,
18,
20
C3:
15,
23,
27,
29 ,
30
C4:
7,
14,
19,
25,
28
C5:
ii,
13,
21,
22,
26
denote the union of these classes,
forms a difference set with parameters
it can be verified that
v,k,Z = 31,15,7.
This is a
special case of Theorem 5.23 given below°
Note that if a union of index classes of
the
v = Nf + i
N TM power residues of some odd prime
it is immediate that all the N th power residue if Nthpower for
f
f
N th power residues are multipliers.
is even (in fact, all solutions of
xf ~ i
Since
-I
modulo
is an v
are
residues) such a union of index classes can only form a difference set odd (see Theorem 3.3 above).
Lemma 5.22.
is odd and
N
Thus:
A union of index classes
power residues of some odd prime
(with or without
v = Nf + i
Ci+ s + Cj+ s + ... + Cm+ s
added) of the
N th
can form a difference set only if
C. + C. + o-o + C m G m for any
s;
f
forms a difference set then
this second difference set can be
obtained from the first by multiplying b y any element of to the first.
0
is even°
If the union of index classes so does
form a difference set, then
Cs
and thus is equivalent
(This fact considerably shortens any exhaustive examination of index
128
class unions.) Let
N = 2,
then with
k < v/2,
as usual, only
CO
need be considered and
this leads to the quadratic residue difference sets of paragraph V.B. N = 4
(with
k < v/2)
only
C0, C 0 + O, C O + C 2
and
CO + C 1
For
need be considered.
The first three of these correspond to the biquadratic residues, the biquadratic residues plus
0
and the quadratic residues;
which of course constitute difference
sets under the conditions given in Theorem 5.18 above.
The cyclotomic numbers
discussed in section V.B. will be used to rule out the remaining possibility. In general,
congruence 5.20 shows that the cyclotomic number
(~,m)N
is the
number of solutions of
- ~ -= 1
for
~
in
C~
and
~
in
Cm.
Thus
(mod
(~-
s, m -
v)
is the number of solutions
S)N
of
g
Nx+~-s
g ~+m-sNv =- i
(rood v)
and therefore of
g
for fixed
d
in
C . s
Nx+~
-
gNy+m
(rood v)
md
Thus, the number of solutions of
- B ~ d
for fixed
d
index classes d
in
C
s
is
in C
C
s
with
. z l' "°'Cz h'
in
CI, G
(mod
in
Cm
is
v)
(~ - s, m - S)N.
(5.3O)
So, for
the nt~nber of solutions of congruence 5.30 for fixed
129
h Js =
h
~
I
i=l
j=l
Thus a difference set corresponds to
(zi - s, zj
C
+ ... + C zI
J0 = J1 . . . . . so only
JN-1 = %°
J0 . . . . .
Since
f
J(N/2)-l = %
Now consider
CO + C 1
(5.31)
S)N
if and only if zh
is odd,
Ji = Ji+N/2
(by equation (5.22))
need be checked.
for
N = 4,
then
Jo = (0'0)4 + (1'°)4 + (°'1)4 + (i'1)4 = (4v - 12 - 8x)/!6 Jl = (-i''i)4 + (0''1)4 + (-1'0)4 + (0'0)4 = (4v-12 + 8x)/16
as the values of x = 0;
but
CO + C1
N/2
given in section V.B. show.
v ( = y 2 + 4x 2)
is
0 - d
Ci
thus
in the union for which
Theorem 5.23. v = 6f + 1
Js i ~ s
Since
f
is odd,
and
C3
should be increased by modulo
belongs to class 1
for each index
N/2. N = 6
v - 3
modulo
4
or
(2)
for an appropriate choice of primitive root 4x 2 + 27.
and
established:
the index classes g
of
v
whenever
g
puts the residue
3
in
C1. )
N = l0 the same technique was applied by Hayashi (1965) yielding:
A set of residues forming a non-trivial difference set modulo
v = 10f + 1
v
The only possibilities are equivalent to one of the
(The appropriate primitive root
Theorem 5.24. a prime
-1
which includes the sextic residues as multipliers may consist
is representable as
For
for which
A set of residues forming a non-trivial difference set modulo
of (1) the qudratic residues for
above.
v
only if
N = 4.
Hall (1956) applied this technique to the case
CO, C 1
J0 = J1
then, so there is no prime
must be counted also.
(congruence 5.25 above);
a prime
Thus
0 is to be added to the index class union~ differences of the type
and
class
y2
form a difference set with
When d - 0
(~,m)4
which includes the
l0 th
power residues as multipliers may
13o consist of classes
(1) the quadratic
CO
and
Yamamoto
CI
index classes
His test has the advantage
5.25.
Let
residues° Let or
g = ii
0
N.
when
added)
be prime with
fth power residues
be the set of all residues
or
i
and define D
or
(2)
the index
v = 31o s
such
forms a difference
The initial form of his result
E
Then the set
4
D = E
of
is a difference
of
N v,
D = E
even and
v)
with
f
odd°
exactly
such that
0
of the
is:
containing
a (mod
N th
set°
that it does not require the prior determination
Let
d = 0 I.
root
modulo
a test for deciding whether a union of
v = Nf + i
be a subset of the set of
v - 3
(with or without
numbers of order
Theorem
for
for the primitive
(1967) developed
power residue
cyclotomic
residues
a
f
Let s
B such
is in
added according
as
Bo d = 0
set if and only if
s(sf + 2d - l) ~ 0
(mod
( jf
if)
N)
(5.32)
(rood v)
(5.33)
i=O
for of
j = 2,4, ~o.,N - 2,
where
Kr
is the sum of the
r
th
powers of the elements
Bo After modifying
N = 4,6,8,10,12
Theorem
congruence
5.26.
A set of residues
v = Nf + i
residues
as multipliers,
residues
(with or without
for
v = 4x 2 + 27
forming a non-trivial
N = 4~6,8,10
condition,
non-trivial
residue
or
12,
0),
the octic residues
or the special
31, 6, I Yamamoto
and, b y way of example, or modified
residue
difference
which includes
m a y consist of the quadratic
In a further attack on the problem, different
applied this test for
w i t h the result:
a prime
sets for
5.33, Yamamoto
residues,
the
set modulo
N th power
the biquadratic
(with or without set of Theorem (1969) develops
0),
the Hall
5.24. a slightly
uses it to show that there exist no
difference
sets for
N = 6,10,14.
[l.e.,
131
he reestablishes parts
(vi), (vii),
(ix) of Theorem 5.18, section V.Bo above.]
In addition to the above facts it is known (Baumert and Fredricksen, that there are
6
index class unions of the
which lead to inequivalent
v,k,h = 127,63,31
18 TM power residues for difference sets.
For
v = 127 N = 6
(1965) developed the requisite prime power cyclotomy and generalized his difference sets to prime powers
v.
However,
W. H. Mills], the only prime powers
4x 2 + 27
sets,
v = 4x 2 + 27
as is shown there
Hall
v = 4x 2 + 27
[proof due to
are in fact primes.
These difference
of Hall, were examined from the point of view of the multi-
plier problem and it was found that indeed every divisor of these sets.
1967)
n
was a multiplier for
[Emma Lehmer (1955 A) gives a proof of this fact and attributes an
earlier proof by means of cubic reciprocity to Hall.]
D.
Generalized Cyclotomy and Difference Sets In 1958 Stanton and Sprott (1958) published a generalization of the following
result:
Theorem 5.27 . and
p + 2
Let
g
are both prime.
be a primitive root of both
p
and
p + 2,
where
p
Then the numbers
2 1, g,g2,...,g(p - 3 ) / 2
0, p + 2, 2(p + 2),...,
form a difference set with parameters
(p- 1)(p + 2)
v,k,h = p(p + 2), (v-i)/2,
(v-3)/4,
i.e.,
a Hadamard difference set. These difference sets (the so-called twin prime sets) were in fact already known, although in slightly different guise.
They had been independently discovered
not only by Stanton and Sprott but also by Kesava Menon (1962), Brauer (1953), Chowla (1945), perhaps first by Gruner (1939) and probably a few others, as they seem to belong to that special class of mathematical objects which are prone to independent rediscovery. Motivated by Theorem 5.27, Whiteman the
k = d + p
numbers
(1962) investigated the problem of when
132
l,g, g2
consistute
for
g
a difference
, .... g
k=
a con~non primitive
root
defined b y
condition
odd primes
Whiteman
p
and
q. Since
Whiteman
showed:
let
x
constitute v
N = 2
(Theorem
and dN
akin to a primitive
and
d
Theorem) is
it follows that set to exist;
cyclotomy
this
for
root must be established.
root of both primes
dN = (p - 1)(q - 1).
and
p
Then there exists an
integers
(s = O,1,...,d
a reduced residue
(5.35)
5.27).
a generalized
be a fixed common primitive
such that the
gSxi
to
g
N = (p - l, q - l)
integer
for such a difference
this problem b y developing
In order to do this something
q;
(p - l, q - l) = N
k = d + p = (v - 1)/N,
condition
v = pq.
Let
~ = (v- I - N ) / N 2
Here
sufficient when
approached
Len~na 5.28.
(5.34)
(easily provided by the Chinese Remainder
is a necessary
is3 in fact,
O,q,2q, .... (p- l)q
(v- l)/N,
(p - l)(q - i) = dN.
q = (N - 1)p + 2
;
set with parameters
v = ~,
of the distinct
d-i
- l;
system modulo
i = %l,...,N
v = pq.
- l)
(5.36)
[That is, all residues prime
are of this form.]
Proof.
Let
x, y
be a pair of integers
satisfying
the simultaneous
con-
gruences
x =- g (mod p),
y - i (mod p)
x -= i (rood q),
y m g (mod q) .
(5.37)
The existence
and uniqueness
of such
x~ y
are guaranteed by the Chinese
133
Remainder g
Theorem.
modulo
v
Note that
i = j.
s # t,
gSxi m gtxJ,
T > 0, ~ > 0. ~;
Note further that the order of
p - l, q - l;
N
gSxi m gtxJ
divides
T.
i.e., it is
contrary to the lemma's assertion.
Since
g
can be written
By (5.37) this implies that so
of
v.
while that fact that the order of
So the assumption
divides
modulo
is the least common multiple
Now assume that shows that
xy m g
p - 1
is
Then (5.37)
rules out the case
x T m g~ m (xy) ~
divides
0 < ~ < N
d
do
~ - T
with
and that
this is a contradiction
q - 1 and the
lemma has been established° [Note that
x
is not unique,
In Whiteman's numbers
generalized
for
y
obviously
cyclotomy the index class
and the generalized i
consists of the
(i,J)N
v)
d
s, t
s i
(5.38)
is the number of members of index
which are followed by members of index class
g x
j.
That is,
(i,J)N
is
of the congruence
+ i -= gtxJ
(mod
v)
(5.39)
0 < s, t < d - io Certain elementary properties
difference
set applications
q - 1 = Nf', d = Nff' f, f'
(mod
c7clotomic number
the number of solutions
where
i
(s = 0,1~°.o,d - l)
a m gSxi
class
serves equally as well.]
below.
of this generalized N
is, of course,
cyclotomy are needed for the even and
for some relatively prime integers
f,f'
p - 1 = Nf, (in particular
are not both even).
x
N
=
g~
-i ---
(mod v)
I
gd/2
for some (mod v)
~ ~ l, 0 _< ~ < d - 1 when
ff'
(5.40)
is odd (5.41)
gW xN/2 (mod v)
when
ff'
is even
i34
where
v
is some fixed integer,
(i,J)N = (i',J')N
0 < v < d - i.
when
r
(J
+
N/2, i
(5.42)
(rood N)
i ~ i', j ~ j'
+
N/2) N
ff'
even
(5.43)
(i,J)N = (N - i, j - i)N = J
2;
1
(6.1)
q-i
see section V.A. for a discussion
Singer noted that when
h = i
there seemed to be only
one equivalence class of difference sets with these parameters and he conjectured that this was in fact the case.
This conjecture of Singer is still open.
(1956, p. 984) has verified Singer's conjecture for 27,32.
Hall
q = 2,3,4,5,7,8,9,11,13,16,25 ,
Beyond this work of Hall apparently very little has been done on this
problem. Strangely enough the largest known class of multiple inequivalent difference sets also have parameters given by (6.1). composite, thus
h > i.
$ > 0,
by (6.1) for which there exist at least k ~ i00, k < v/2,
as follows:
2
121, 40, 13
and
is necessarily
for
2
for
127, 63, 31.
For while those parameters
~
there exist values of
v, k, k
given
pairwise inequivalent difference sets.
the known multiple inequivalent difference sets are
for 31, 15, 7; 6
N + i
In fact (see section V.Ao for details) Gordon, Mills and
Welch (1962) have shown that for any
For
Here, though,
43, 21, i0;
2
for
63, 31, 15;
4
for
[There may well be others in this range.
v, k, h (k ~ i00)
which have associated cyclic
difference sets have been determined (see section VI.B) not much beyond the work of Hall (1956, limited to
k ~ 50)
is known about multiple inequivalent difference
i44
sets in this range.] Two block designs with the same parameters are considered equivalent if there exists two permutations (generally different), one acting on the objects and the other acting on the blocks, which take the one design into the other.
On the other
hand, equivalence of difference sets (see section I.B) is a more restrictive relation.
Thus, it could happen that two cyclic difference sets were inequivalent
while their associated block designs were not. this behavior.
There are no known examples of
In particular, it does not happen for the parameter sets
mentioned just above.
(k ~ 100)
In these cases, inequivalence of the associated block
designs follows from the distribution of sizes of the intersections of triples of blocks.
For example, the distribution of intersection size for all block triples
containing a particular fixed block is 420 of size of the
v,k,~ = 31,15,7
difference sets.
set has such block triples intersections of size
4.
3
and
15 of size
Whereas the other 90 of size
2,
195
7
31, 15, 7 of size
for one difference
3
and
150
Since the distributions of block intersection sizes must be the same
for equivalent block designs, it is clear that these designs are not equivalent. The question of whether the multiple inequivalent difference sets of Gordon, Mills and Welch necessarily lead to inequivalent block designs is open. [While no examples of inequivalent cyclic difference sets generating the same block design are known, there are examples of block designs which are generated by more than one difference set. these examples is non-cyclic.
Of course, at least one of the difference sets in
In fact, Bruck (1955 , p. 475) has shown that the
block design associated with a cyclic difference set with parameters n
2
+ n + i, n + i, i
n ~ i
B.
modulo
3
(these are only known to exist for prime power
v,k, h = n)
for
can also be generated by a non-Abelian group difference set.]
Searches In 1956 Marshall Hall (1956) published the results of his search for difference
sets having
k < 50.
v,k,h(k ~ 50),
With but
12
exceptions, he decided, for each parameter set
whether or not a difference set existed.
When the difference set
145
was not a member of one of the families of Chapter he listed the residues modulo
v.
V
(and in some other cases)
For many of the smaller parameter sets he decided
whether or not multiple inequivalent difference sets existed (see section VI.A. above ). Hall's twelve undecided cases were all resolved negatively.
In fact many of
the existence tests presented in section II.E. above were developed specifically for the purpose of deciding Hall's twelve cases. Baumert (1969) extended Hall's search to exactly
74
parameter sets
sets existed.
These
difference sets;
74
Utilizing these powerful tests
k < i00o
v,k, h(k _~ I00)
He found that there were
for which non-trivial cyclic difference
parameter sets have associated with them
85
known
there may be more, since Baumert made no attempt to find multiple
inequivalent difference sets beyond those already known. below contains all
85
known difference sets for which
Table 6.1 of section VI.Do k < i00.
As reported in more detail in section IV.A., Hall (1947) checked that, for n _~ I00,
all planar
(i.eo,
h = i)
difference sets had
n
a prime power.
This
conjecture, which is still outstanding, was checked further by Evans and Mann (1951) up to
n < 1600.
Dembowski (1968, p. 209) states that V. Ho Keiser (un-
published) has checked it to
n _~ 3600.
Hadamard difference sets
(i.e., those with
have been searched for, through
v < i000.
v,k,h = 4t - i, 2t - i, t - i)
The results of these searches (Golomb,
Thoene, Baumert) are that, except for possible additional multiple inequivalent difference sets, the only unknown Hademard difference sets possible, have
v = 399, 495, 627, 651, 783
or
975°
v < 100%
(See section IV.B. for a discussion of
the known Hadamard difference sets. ) The difference sets associated with circulant Hadamard matrices (see section IVoC. ) have been sought.
These have parameters
v = 4N 2, n = N 2 .
Turyn (1968)
surveys the results in this area, most' of which are due to him, and shows that, except for
v = 4,
none such exist with
v < 12,100.
lh6
C.
Some Examples The general tenor of these examples seems to be that, at least from an alge-
braic number theoretic point of view, difference sets are no better behaved than they absolutely have to be. Consider the quadratic residue difference set with parameters 25.
Here
modulo
n = 26,
103
is
are minimal.
the order of
17,
i.e.,
2
modulo
103
251 e l, 1317 ~ 1
Furthermore, no power of
13
is modulo
51
and the order of 103,
K({103)
defined by
the automorphism 8 ~i03 -+ ~i03'
T,
P's
hand, Theorem 5.20 shows that set.
13
~i03 -+ ~i03'
defined by
fixes both
2
or and
4
modulo
13
in the
103.
(13) : PiP2P3PiP2P 3
where the bar denotes complex conjugation. ~,
2
as:
(2) : ~ ,
automorphism
13
where the exponents
is congruent to
Thus Theorem 2.19 gives the prime ideal factorizations of cyclotomic field
v,k, ~ = 103,51,
and 2
Q's, 13
248 -= 13
it follows that the
fixes all these prime ideals;
2 4103,
~i03
and
Since
since
fixes only the
Q's.
1316 ~ 8 (mod 103).
But
whereas T 3,
i.eo,
On the other
are both multipliers of this difference
So, without loss of generality, the prime ideal decomposition of the ideal
(G(~I03))
is
(e(~lO3)) = QP~P33
where the action of the multiplier according to the multiplier
PI ~ P2 ~ P3 ~ PI" 2,
2
permutes the prime ideal divisors of
That is, while the ideal
(0(~I03))
13
is fixed by
its individual prime ideal divisors are not.
Consider the planar difference set whose parameters are consists of the residues
i, 6, 7, 9, 19, 38, 42, 49
notation of section III.D.,
modulo
v,k, h = 57,8,1. 57.
Using the
It
i~7
@[57](x) : x + X 6 + x 7 + x 9 + x !9 + x 38 + x 42 + x 49
e[19](x ) = 2 + x + x 4 + x 6 + x 7 + x 9 + x II
and the prime ideal divisors of
are (see Theorem 2.19):
(@(~19))
2
A : (7, ~ 9 + 6~I 9 + 3~19 + 6) 2 + 5~i 9 + 6)
C = (7, ~ 9
while
A, B, C
ideals over
divide
K({57).
which lie above
A
(@(~i~)). Let
and
al, a 2 B
2 + 4~i 9 + 4~i 9 + 6)
Each of these six ideals splits into two prime and
bl, b 2
respectively.
be the prime ideals of
Then
(e(~7)) ~ ala2hlb2ClC 2
where
2 + 2~57 + 6) a I = (7, ~ 57 + ~57
~2 = (7, ~ 7
2 + 6~57 + 5~57 + 6)
b I = (7, ~7 + 3 ~57 2 + 6~57 + 6) 2
~2 = (7, ~ 7
+ 2~57 + 2~57
c I = (7, ~ 7
+ ~57 + 6)
c 2 = (7, ~ 7
+ 4~57 + 6).
+ 6)
K({57)
148
Thus, the tempting assumption that the fact that
A
divides the ideal
(e(~19))
implies that one of
2 2 al, a2, ala 2
(e(~57))
Besides certain theoretical implications, this complicates the
is wrong.
necessarily divides the corresponding ideal
application of the constructive existence test of section III.D. Consider the planar difference set v, k, h K(~73 )
are
73, 9, l;
n = 8.
[1,2,4,8,16,32,37,55,64)
whose parameters
Now one of the prime ideal divisors of
(2) in
is
6 2 A = (2, ~ 3 + ~73 + ~ 3 + ~73 + I)
and
A~
divides
(e(~73)).
Thus, the simplifying assumption, that one and only
one of a complex conjugate pair of prime ideal divisors of
n
divides
(e(~))
is
not correct in general°
D.
A Table of Difference Sets Table 6.1 below contains all
85
known difference sets for which
k ~ i00.
As pointed out earlier, there well may be others; for the question of the existence of multiple inequivalent difference sets has not been solved for all of the parameter sets
v,k, ~ (k ~ i00)
has been shown that these
74
having associated difference sets.
74
However, it
parameter sets are the only ones that need be
considered. Each difference set is identified by are the prime divisors of
n)
v, k, h, n, pl , P2' P3
(where the
Pi
and by a class indicator which indicates the family
or sub-family to which it belongs.
These are:
SN
-
Hyperplanes in projective
N
L
-
Quadratic Residues (Theorem 5.15)
B
-
Biquadratic Residues (Theorem 5.16)
BO
-
Biquadratic Residues and
0
-
0ctic Residues (Theorem 5.18)
H
-
Hall's Sets (Theorem 5.23)
O
space (Theorem 5.1)
(Theorem 5.±8)
149
TP GMW
-
Twin Prime Sets (Theor~n 5.27) Gordon, Mills, Welch (Section V.A)
W
-
Whiteman Set (Theorem 5.31)
*
-
not one of the above .
Where multiple inequivalent sets are known, they are distinguished by etc., which is added to the parameter v,k,k = 31,15~7 respectively.
v.
Thus the two inequivalent
difference sets are designated
31A, 15, 7
and
31B~ 15, 7
The table is sorted in order of increasing values of
k.
A, B, C
150
{.0 I,.F~
CO
0
,,< II __I O,J U ~
II OJ U)
O_ (~ .J
~ ~
,{"J F,,/')
(~j (~J (I)
~
._~,
~
~ ~
(,r)
IX) (',~ 0..I 6r') ....-,~
.....,I oO
L~ b.J L~ I..L
0 .,~
..---~ ('~o
Z
::>
~ ~
f'-.
~
~
~
(Y)
..,,i"
-"4 .-~
~ 03
r,"~
~D ~
0'~
0~I
(T:, 0-,, (~r) ,LC'J (k~ ('~ ~.~C)~ r ['~ ...~ r-- 0",. L~) ,wC~I"- 0
{'~J oO [vJ 0"~ OJ -.'t~ ~
OJ ..i~ ('~, (Y~, f,') -,,-r .-,-,-,-,-,-,-,-,~r-.,. ~ } .--i
