Binary Codes of Strongly Regular Graphs

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics Designs, Codes and Cryptography, 17, 187–209 (1999) c 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Binary Codes of Strongly Regular Graphs WILLEM H. HAEMERS Department of Econometrics, Tilburg University, 5000 LE Tilburg, The Netherlands RENE´ PEETERS Department of Econometrics, Tilburg University, Brabant 5000 LE Tilburg, The Netherlands JEROEN M. VAN RIJCKEVORSEL Department of Econometrics, Tilburg University, Brabant 5000 LE Tilburg, The Netherlands Dedicated to the memory of E. F. Assmus Received May 19, 1998; Revised December 22, 1998; Accepted December 29, 1998 Abstract. For strongly regular graphs with adjacency matrix A, we look at the binary codes generated by A and A + I . We determine these codes for some families of graphs, we pay attention to the relation between the codes of switching equivalent graphs and, with the exception of two parameter sets, we generate by computer the codes of all known strongly regular graphs on fewer than 45 vertices. Keywords: binary codes, strongly regular graphs, regular two-graphs 1. Introduction Codes generated by the incidence matrix of combinatorial designs and related structures have been studied rather extensively. This interplay between codes and designs has provided several useful and interesting results. The best reference for this is the book by Assmus and Key [3] (see also the update [4]). Codes generated by the adjacency matrix of a graph did get much less attention. Especially for strongly regular graphs (for short: SRG’s) there is much analogy with designs and therefore similar results may be expected. Concerning the dimension of these codes, that is, the p-rank of SRG’s, several results are known: see [5], [17]. In addition, it has turned out that some special SRG’s generate nice codes, see [12] and [22]. The present paper is a first step to a more structural approach. We restrict to binary codes, not only because it is the obvious case to start with, but also since for the binary case there is a relation with regular two-graphs and Seidel switching that has already proved to be useful: see [12] and [7]. The paper is organised as follows. First some conclusions about the code are presented that are based upon the parameters of the SRG only. Often, but not always, the dimension follows from the parameters, but only in a few obvious cases the code itself is determined by the parameters. Next we pay attention to codes of some famous SRG’s. This includes the Paley graphs, the graphs related to a symplectic form over F2, the Latin square graphs and the block graphs of Steiner 2-designs. Several good codes appear. For example the 188 HAEMERS, PEETERS AND VAN RIJCKEVORSEL quadratic residue and simplex codes. In Section 5 we study the relation between Seidel switching in a graph and the generated code. Here we introduce the concept of a two- graph code, which unifies in a sense the codes of SRG’s that are equivalent under Seidel switching. This section contains a useful result (Theorem 5.1), that relates the presence of the all-one vector to the dimension of the code. It implies for example that the code of an SRG with parameters (36,14,4,6) contains the all-one vector. In Section 6 we present the computer results. Here we give the weight enumeraters of the codes (and two-graph codes) of the known SRG’s (and regular two-graphs) on fewer than 45 vertices. For two parameter sets ((35,16,6,8) and (36,15,6,6)) there are too many SRG’s known to present all weight enumerators, so we restrict to the ones from Steiner triple systems and Latin squares. In some cases we find interesting codes. For example there is a Latin square of order 6 that generates an optimal code of length 36, dimension 13 with minimum weight 12 (see Section 6.3). We also find out that for several parameter sets the non-isomorphic graphs produce non-isomorphic codes. But there are interesting exceptions. For example the 180 known SRG’s with parameters (36,14,4,6) together generate only 76 non-isomorphic codes. The reader is assumed to be familiar with the basis concepts of codes and designs, which can for example be found in [3] or [10]. 2. Preliminaries For an integral n v matrix A we define the binary code CA of A to be the subspace of = v V F2 generated by the rows of A (mod 2). We start with some known lemmas for symmetric integral matrices (see [5], [6] or [17]). LEMMA 2.1 If A is a symmetric integral matrix with zero diagonal, then 2-rank(A) (i.e. the dimension of CA) is even. Proof. Let A0 be a non-singular principal submatrix of A with the same 2-rank as A.Over Z, any skew symmetric matrix of odd order has determinant 0 (since det(A) =det(A>)). Reduction mod 2 shows that A0 has even order. LEMMA 2.2 If A is a symmetric binary matrix, then diag(A) ∈ CA. P P ∈ C⊥ ( ) = ( ) = > = Proof. Suppose x A . Then i A iixi i, j A ijxi xj x Ax 0 (mod 2), so ⊥ ( ) ( ) ⊥ C⊥ x diag A . Hence diag A A . With these lemmas we easily find a relation between the codes CA and CA+J (J is the all-one matrix, 1 is the all-one vector). PROPOSITION 2.1 Suppose A is the adjacency matrix of a graph then CA CA+J and the following are equivalent: i. CA = CA+J , ii. 1 ∈ CA, iii. dim(CA+J ) is even. BINARY CODES 189 Proof. By Lemma 2.2, diag(A + J) = 1 ∈ CA+J ,soCA+J = CA +h1i and the equivalence of i and ii follows. By Lemma 2.1 we have 2-rank(A) is even and so i ⇔ iii. The next proposition gives a trivial but useful relation between CA and CA+I . C⊥ C PROPOSITION 2.2 If A is a symmetric integral matrix, then A A+I with equality if and only if A(A + I ) = 0 (mod 2). ∈ C⊥ = ( + ) = ∈ C Proof. Suppose x A . Then Ax 0 (mod 2), so A I x x and hence x A+I . ( + ) = C C⊥ Clearly A A I 0 (mod 2) reflects that A+I A , which completes the proof. A strongly regular graph with parameters (v, k,,), for short SRG(v, k,,),isa regular graph 0 with v vertices, valency k (1 k v 2), and the number of common neighbours of two distinct vertices x and y is equal to or respectively, depending on whether x and y are adjacent or not. Let 0 be a SRG(v, k,,). Clearly 0, the complement of 0,isaSRG(v, v k 1,v 2k + 2,v 2k + ) and moreover, the adjacency matrix A of 0 satisfies A2 = kI + A + A, (1) where A = J I A is the adjacency matrix of 0. The valency k is an eigenvalue of A with eigenvector 1. It follows that A has just two more eigenvalues, say r and s (r > s), with multiplicities f and g respectively, which satisfy: rs = k , r + s = , (k r)(k s) = v, v = f + g + 1 and 0 = k + fr + gs. √ Moreover, either r and s are integers or r =1s = (1+ v)/2, f = g = k = (v1)/2 and = +1 = k/2. If 0 or 0 is a disjoint union of complete graphs, 0 is called imprimitive and the codes CA and CA+I are trivial. Throughout, we will assume that 0 is primitive (not imprimitive). If = , Equation (1) becomes A2 = (k )I + J, that is, A is the incidence matrix of symmetric 2-(v, k,)design and 0 is called a (v, k,)graph. Similarly, if +2 = , A+ I represents a 2-(v, k + 1,+ 2) design. It may happen, however, that two non-isomorphic (v, k,)graphs, 01 and 02 with adjacency matrices A1 and A2 say, give isomorphic designs. If this is the case then A1 = PA2 for some permutation matrix P and it is obvious that C = C C = C + A1 A2 and A1+J A2+J . Similarly, if A1 and A2 I represent the same design, = ( + ) C = C then A1 P A2 I and then A1 A2+I . The standard example is given by the two SRG(16, 6, 2, 2)’s (the lattice graph L(4) and the Shrikhande graph) and the unique SRG(16, 5, 0, 2) (the Clebsch graph). The three graphs produce the same symmetric 2- (16, 6, 2) design, and therefore the same code (take CA+I for the Clebsch graph). We refer to [10] for these and other results on strongly regular graphs and designs. 190 HAEMERS, PEETERS AND VAN RIJCKEVORSEL 3. Facts from the Parameters Here we present some properties of the binary codes of a strongly regular graph 0, using only the parameters of 0. PROPOSITION 3.1 Suppose 0 has non-integral eigenvalues. ⊥ i. If is odd (i.e. v = 5 mod 8) then CA = 1 and CA+I = V. v = C⊥ = C (C ) = (C ) = ii. If is even ( 1 mod 8) then A A+I and dim A dim A+I 1 2 (= f = g = k = (v 1)/2). Proof. If is odd, Equation (1) becomes A2 = A + I + J (mod 2), so (A + J)(A + I ) = I ⊥ 2 (mod 2), hence CA+J = CA+I = V and CA = 1 . Suppose is even. Then A = A (mod 2) C = C⊥ so A+I A .

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