A 2-(22, 8, 4) Design Cannot Have a 2-(10, 4, 4) Subdesign

Designs, Codes and Cryptography, 27, 257–260, 2002  C 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

A 2-(22, 8, 4) Design Cannot Have a 2-(10, 4, 4) Subdesign ∗ ¨ ˚ PATRIC R. J. OSTERG ARD [email protected] Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015 HUT, Finland

Communicated by: P. Wild Received November 16, 2000; Revised September 5, 2001; Accepted October 5, 2001 Abstract. The smallest BIBD, as for the number of points and blocks, whose existence is still undecided is 2-(22, 8, 4). Possible subconfigurations of such a design, namely 2-(10, 4, 4) designs, are here ruled out. The result is obtained by classifying all 2-(10, 4, 4) designs and trying to find 2-(22, 8, 4) designs by solving instances of the maximum clique problem. Keywords: BIBD, codes of designs, maximum clique problem Mathematics Subject Classification: 05B05

1.

Introduction

A t-(v, k, λ) design is a family of k-subsets, called blocks, of a v-set of points such that each t-subset of the v-set is contained in exactly λ blocks. A 2-(v, k, λ) design is called a balanced incomplete block design (BIBD). Two more parameters that are related to a design are b, the number of blocks, and r , the number of blocks in which a point occurs. An incidence matrix of a design is a v × b (0,1)-matrix with the rows indexed by the points and the columns indexed by the blocks, where a 1 (0) indicates that a block contains (does not contain) a point. The values of b and r can easily be determined from the values of the other parameters, as vr = bk

and r (k − 1) = λ(v − 1).

The smallest BIBD, as for the number of points and blocks, whose existence is still undecided is 2-(22, 8, 4). In fact, this has since the early days of design theory been the smallest undecided case. Along the years many results on this problem have been presented; see [11] for an extensive survey. People have studied properties such a design can or must have and some of these cases have been eliminated. This work, to which the present paper contributes, converges slowly towards a final solution. ∗ Supported

by the Academy of Finland under grants 44517 and 100500.

¨ ˚ OSTERG ARD

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In Section 2, we discuss configurations that could occur as subdesigns of a 2-(22, 8, 4) design. In particular, we classify the 13,769,944 2-(10, 4, 4) designs and try to extend these to 2-(22, 8, 4) designs by solving instances of the maximum clique problem. The results are presented in Section 3: no 2-(22, 8, 4) designs are produced in this way, and hence such a design cannot have a 2-(10, 4, 4) subdesign. This further implies that the dual code of the code of points of a 2-(22, 8, 4) design cannot have a word of weight 3. Several years of cpu time were consumed to settle this case.

2.

Codes, Designs, and Graphs

Coding theory plays a central role in our approach, which is due to Bate et al. [2] (and which is also surveyed in [11]); coding theory had earlier been applied to other similar design-theoretical problems in [3, 9]. Both the columns and the rows of the incidence matrix of a design generate (here, binary) codes; these codes are called the code of blocks (or just code), denoted by C (the term block code is sometimes used but has another well-known meaning in coding theory), and the code of points (or point code), denoted by D, respectively [1, 10]. The spaces in which these codes reside are F2v and F2b , respectively, and they have the same dimension. From the parameters of a 2-(22, 8, 4) design, it is not difficult to deduce that the weights of all words in its code of points must be divisible by four, and the code is then selforthogonal: D ⊆ D ⊥ , and so dim(D) ≤ dim(D ⊥ ). As dim(D) + dim(D ⊥ ) = 33, we get dim(C) = dim(D) ≤ 16 and so dim(C ⊥ ) ≥ 22 − 16 = 6. The result dim(C ⊥ ) ≥ 6 means that there are at least 26 sets of rows of the incidence matrix whose columns sum to zero, that is, they have an even number of 1s. One such set consists of all rows and corresponds to the all-one codeword, but what other sets could we have? The blocks of such subdesigns have even weight. Moreover, the number of rows is greater than 2 as no two rows of the incidence matrix coincide. The subdesigns are therefore even pairwise balanced designs (PBDs) on s > 2 points. If such a PBD has bi blocks of size i, we get    bi = b, ibi = r s, i(i − 1)bi = λs(s − 1), and the admissible solutions, called B-vectors, are given in Table 1. The solutions in the table are restricted to s ≤ 11 as the subdesigns come in pairs due to the all-one vector in C ⊥. Types 4 and 5 can be eliminated directly [2], and we are left with ten cases. Two of the PBDs, types 1 and 8, are BIBDs. Tens of different 2-(10, 4, 4) designs have been checked [2, 5, 11] but none of these turned out to produce a 2-(22, 8, 4) design. We decided to check all 2-(10, 4, 4) designs. The 2-(10, 4, 4) designs have recently been enumerated [4]. There are 13,769,944 such designs; this number served as a partial correctness check when we produced these designs. For each 2-(10, 4, 4) design, we add three all-zero columns to its incidence matrix and construct a graph with a vertex for each vector of weight 12 that has four common 1s with

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A2-(22, 8, 4) DESIGN CANNOT HAVE A 2-(10, 4, 4) SUBDESIGN Table 1. B-vectors with s ≤ 11. Type

b0

b2

b4

b6

b8

s

1 2 3 (4) (5) 6 7 8 9 10 11 12

9 3 2 1 0 1 0 3 2 1 0 0

24 24 27 30 32 16 19 0 3 6 9 8

0 6 3 0 0 16 13 30 27 24 21 24

0 0 1 2 0 0 1 0 1 2 3 0

0 0 0 0 1 0 0 0 0 0 0 1

4 6 6 6 6 8 8 10 10 10 10 10

each row of that matrix. Edges are inserted between vertices in the graph that correspond to vectors with four common 1s. Now there exists a 2-(22, 8, 4) design with the given 2-(10, 4, 4) embedded if and only if a maximum clique of the graph is of size 12. 3.

The Result

The algorithm developed by the author and presented in [8] played a central role in the computation of the sizes of maximum cliques. The computation was distributed over several computers in a local network using the batch system autoson [6]. It was completed using 6 years of cpu time on 66–500 MHz PC computers, equivalent to approximately 3.5 years of 500 MHz PC cpu time (that is, on average 8 seconds per instance on a 500 MHz PC). Fourteen of the graphs, for certain 2-(10, 4, 4) designs with (often large) nontrivial automorphism groups, have more than 10,000 vertices (with as many as 29,970 vertices in the largest graphs); these had to be considered separately by adding a few points and rejecting isomorphs before feeding them into the clique program. The result is that no 2-(22, 8, 4) design can be constructed in this way. In Table 2 we give the sizes of the maximum cliques and how many times they appear. It has earlier been known that there is a multitude of configurations that are in some sense very close to being Table 2. Distribution of maximum clique sizes. Maximum Clique

Count

5 6 7 8 9 10

58 9,051 3 43 33,016 13,727,773

Total

13,769,944

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¨ ˚ OSTERG ARD

a 2-(22, 8, 4) design [7]. Therefore the great number of instances when the maximum clique is of size 10 is perhaps not surprising (we cannot have a maximum clique of size 11, so this is as close as one can get). The main result of this paper has one further implication: D ⊥ , the dual of the code of points, cannot have a word of weight 3. Such a word of weight 3 would imply three blocks with pairwise intersection in four points. Then there are ten points that do not occur in those three blocks, and it follows by a counting argument that the only possible configuration in these ten points is a 2-(10, 4, 4) design. For a discussion of the weight enumerator of D ⊥ , see [11]. Due to a large number of PBDs corresponding to some of the remaining B-vectors, the amount of cpu time needed to eliminate these, and thereby to settle the existence problem, is discouragingly large. In particular, there is apparently billions of PBDs corresponding to B-vector 11.

Acknowledgment The author would like to thank the anonymous referee for valuable comments.

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