3-Class Association Schemes and Hadamard Matrices of a Certain Block Form

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1 Europ J Combinatorics (1998) 19, Article No ej Class Association Schemes and Hadamard Matrices of a Certain Block Form R W GOLDBACH AND H L CLAASEN We describe 3-class association schemes with adjacency matrices D i such that for an appropriate numbering of the relations, D 0 + D 1 D 2 + D 3 is an Hadamard matrix of the following form: J g H 12 H 1g H 21 J g H 2g, H g1 H g2 J g where g = 2s for natural s, all blocks are of size g, J g is the all-one matrix and H ij J g = J g H ij = 0 Also we construct all Hadamard matrices of this form in which all blocks H ij have rank 1 c 1998 Academic Press 1 PRELIMINARIES For association schemes we use the notation introduced by Delsarte in [5], but see also [11, 12] We use freely the notation given in [12] for non-symmetric 3-schemes and the notation of [17] are used as well For non-symmetric 3-schemes we use the following shorthand notation for the parameters of (X, R): u = v 1 /v 3, u = µ 1 /µ 3 and α = p11 1, β = p2 11, γ = p1 33, δ = p1 13, ɛ = p1 23, λ = p33 3, = P 1(1), = P 3 (1), = P 1 (3), = P 3 (3) In this paper we use the following schemes (1) A symmetric 2-scheme is said to be a group-divisible 2-scheme (of type (g, h)) (it is also called a GD-scheme (of type (g, h)), where g, h N \{0, 1}, if for a suitable numbering of the relations and a suitable numbering of the eigenspaces the first eigenvalue matrix P has the following form: P = ( 1 g 1 ) g(h 1) 1 g 1 g Using P and p 2 11 = 0 one can calculate all the intersection numbers The GD-schemes are the imprimitive symmetric 2-schemes (2) We say that a symmetric 3-scheme is of type L 1,s (2s) if its first eigenvalue matrix has the following form: 1 s(2s 1) s(2s 1) 2s 1 1 s s 1 P = 1 s s 1 1 s s 2s 1 The intersection numbers can now be calculated (cf [2, Theorem II36]) One finds p11 1 = p1 12 = p2 12 = p2 22 = s(s 1) Once one has this, all the intersection numbers follow easily As R 0 R 1 R 2 is an equivalence relation, a 3-scheme of type L 1,s (2s) is imprimitive Symmetric 3-schemes, in general, are discussed in [14] The second author, H L Claasen, died on 26 May /98/ $3000/0 c 1998 Academic Press

2 944 R W Goldbach and H L Claasen (3) The non-symmetric 3-schemes as they are treated in [11, 12] The next lemma is proved in [17, p 280] Although there is the condition that the Hadamard matrix is also symmetric, the symmetry of the matrix is not used in the proof LEMMA 11 If H is a regular Hadamard matrix (H J h = J h H = kj h ) of order h, then h = 4s 2 for some integer s and k = 2s Before stating Definition 13 we note that by Theorem 12 a more general possibility for Hadamard matrices to have a so-called checkered form is excluded The theorem is an easy consequence of Lemma 11 THEOREM 12 that Let g, h N \{0, 1} and let H be a square block matrix of order gh such H 11 H 12 H 1h H 21 H 22 H 2h H = H h1 H h2 H hh where the H ij are square matrices of order g If H is an Hadamard matrix, H ii = J g for all i, H ij J g = J g H ij = 0 for all i and j such that i = j, then g = h DEFINITION 13 Let s N \{0} and let H be a square block matrix of order 4s 2 such that H 11 H 12 H 1g H H = 21 H 22 H 2g, H g1 H g2 H gg where the H ij are square matrices of order g = 2s Then H is called a checkered Hadamard matrix of order 4s 2 if (1) H is an Hadamard matrix, (2) H ii = J 2s for all i, (3) H ij J 2s = J 2s H ij = 0 for all i and j such that i = j A checkered Hadamard matrix H of order 4s 2 such that H I 2s J 2s is skew-symmetric is called a skew-checkered Hadamard matrix If C is a conference matrix such that the non-zero entries in the ( first row ) and the first column 0 j all are +1 (in that case C is said to be normalized) and if C = j T, then S is called the S core of C For any association scheme (X, R) we use v to denote the number of elements of X I t denotes the identity matrix of order t and J t the t t all-one matrix 2 NON-SYMMETRIC 3-SCHEMES AND HADAMARD AND CONFERENCE MATRICES We first investigate the possibilities for Hadamard and conference matrices to be useful for the construction of non-symmetric 3-schemes The proof of the next lemma is straightforward LEMMA 21 Let (X, R) be a non-symmetric 3-scheme and let H = η 0 D 0 + η 1 D 1 + η 2 D 2 + η 3 D 3 with η 0 {0, +1, 1}, η i { 1, +1} for i {1, 2, 3} and η 1 η 2 = 1 We give the following conditions:

3 Then, α β + γ = 0, 2η 0 η 3 + 2u(δ ɛ) + λ = 0 Non-symmetric 3-schemes 945 (1) if η 0 = 0, H is an Hadamard matrix if and only if the intersection numbers of (X, R) satisfy the above set of conditions; and (2) if η 0 = 0, H is a conference matrix if and only if the intersection numbers of (X, R) satisfy the above set of conditions By the preceding lemma in essence only the following possibilities have to be considered (1) For η 0 = 0: H = D 0 + D 1 D 2 + D 3 and H = D 0 + D 1 D 2 D 3 (2) For η 0 = 0: H = D 1 D 2 + D 3 and H = D 1 D 2 D 3 In Lemma 21 we excluded the case η 1 η 2 = 1 because then one considers, in fact, the symmetric closure of (X, R) The conditions given in Lemma 21 do not appear to be easy to tackle Therefore we introduce a new set of parameters for non-symmetric 3-schemes The method is a generalization of the one used by Enomoto and Mena [6] THEOREM 22 Define for a non-symmetric 3-scheme (X, R) the following parameters: ϕ =, ψ = and = u Then ϕ,ψ, Z and ϕ = 0 if and only if = 0 If ϕ = 0, then (2ψ + ψϕ + )(2 + ϕ + 1) v =, v 1 = 2ψ + ψ + + ψϕ( ), v 3 = 2ψϕ + ϕ + µ 1 = v 1 + v 3ψ v 1 ϕ 2ψ ϕ = v 2ψ + ψϕ + 1 (2ψ + ϕ + 1), µ 3 = v 3 2 v 3ψ v 1 ϕ 2ψ ϕ = v 2 + ϕ ψ + ϕ + 1, α = ψ (ψ + 1 ϕ) ( ), β γ = ψϕ(1 + ϕ),δ = ϕ(1 + ψ)( ), ɛ = ψϕ( ), ψϕ(1 + ϕ) λ = ϕ 2ψ 1 +, uγ = ψ(1 + ϕ) ( ), uδ = (1 + ψ)( ), uɛ = ψ( ) The formulas marked with ( ) are also valid in the case ϕ = 0 PROOF ϕ = 0 if and only if = 0 is evident For now we allow ϕ = 0 From Theorem 33 in [12] we find ψϕ(1 + ϕ), u = ϕ, = ψ (ψ ϕ) ( ), α β = 1 ϕ, δ ɛ = ϕ and uγ = ɛ + ψ Substituting this into (α β)ɛ = uγ(ɛ δ) (formula (1) in [12]), one finds ɛ = ϕψ From ϕ = δ ɛ one finds = uϕ = uδ uɛ Z Now it readily follows that uγ = ψ(1 + ϕ) and δ = ϕ(1 + ψ)

4 946 R W Goldbach and H L Claasen If we suppose ϕ = 0, then u = /ϕ and so γ = ϕψ(1 + ϕ)/, uδ = (ψ + 1) and uɛ = ψ (Since δ = ɛ = 0 if and only if ϕ = 0, which, in turn, is equivalent to = 0, we find that the formulas for uδ and uɛ also hold if ϕ = 0) Again from Theorem 33 in [12] we derive v 1 = u(ɛ +δ +γ)= α +β +ɛ, v 3 = ɛ +δ +γ = 1 + 2uγ + λ Hence v 1 = 2ψ + + ψ + ψϕ (also valid if ϕ = 0), v 3 = 2ψϕ + ϕ + ϕψ(1 + ϕ) and λ = 1 2ψ + ϕ + ϕψ(1 + ϕ) (The formulas for v 3 and λ are only valid if ϕ = 0) From α + β = v 1 ɛ and α β = 1 ϕ we derive the formulas for α and β given in the theorem (also valid if ϕ = 0) and from Theorem 33 in [12] we get the formulas for µ 1 and µ 3 The rest of the assertions are now easily derived The integers ϕ, ψ and as defined in Theorem 22 are called the non-standard parameters of (X, R) Throughout this section we shall use the letters ϕ, ψ and in this way Non-standard parameters are considered in [13] In Theorem 22 we had, in general, to exclude the case ϕ = 0 This is, however, not an essential restriction because in the case ϕ = 0, in the terminology of [11], the scheme (X, R) is the splitting of a GD-scheme of type (g, h) according to case II However, this situation has been completely dealt with in [11] It follows from Theorem 22 that once v 1, α and β are known and α β = 1 (ϕ = 0), then all the parameters of (X, R) can be calculated Indeed, for (X, R) the following hold if α β = 1: ϕ = β α 1, ψ = v 1 α β ϕ and = α + β ψ 2ψ + 1 When considering the possibility of the use of Hadamard matrices for the construction of a non-symmetric 3-scheme (X, R) the only cases which are of importance are ϕ = 0 and ϕ = 1 (For more details we refer to [11]) (a) For ϕ = 0((X, R) is the splitting of a GD-scheme of type (g, h) according to case II) it suffices to know that g 3 (mod 4), α β = 1 and γ = g(h 1), hence γ 3 (b) For the case ϕ = 1((X, R) is the splitting of a GD-scheme of type (g, h) according to case IV) we need further knowledge of the parameters We have: v = gh, v 1 = v 2 = 1 2 g(h 1), v 3 = g 1, µ 1 = µ 2 = 1 2h(g 1), µ 3 = h 1, α = β = 1 4 g(h 1), γ = 0, δ = 1 2 (g 2), ɛ = 1 2 g, λ = g 2, ϕ = 1, ψ = 1 2 g, 1) = g(h 2(g 1) We have g h 0 (mod 2) and since Z, one sees that g 1 h 1 (in particular g h) LEMMA 23 Let (X, R) be a non-symmetric 3-scheme Then the following hold (1) D 0 + D 1 D 2 + D 3 is an Hadamard matrix if and only if ϕ = 1 and ψ = In this case g = h and g 0 (mod 2) (2) D 0 + D 1 D 2 D 3 is never an Hadamard matrix

5 Non-symmetric 3-schemes 947 PROOF If ϕ = 0, then α β + γ = 0 if and only if γ = 1 However, as noted earlier for any non-symmetric 3-scheme with ϕ = 0, γ 3 So we can assume ϕ = 0 Since now ϕ = 0, the conditions of Lemma 21 are equivalent to (1 + ϕ)( ψϕ) = 0 and η 0 η ϕ ψ = 0 (1) If η 0 η 3 =+1and ϕ = 1, (1) is equivalent to ψ = But then g = h and g 0 (mod 2) follow easily If η 0 η 3 =+1and ϕ = 1, (1) is equivalent to = ψϕ and (ϕ 1)(ψ + 1) = 2 But (ϕ 1)(ψ + 1) = 2is not possible for any non-symmetric 3-scheme in the present situation because if ϕψ = 0, then ϕ, ψ and have to have the same sign (use the formulas for ɛ and uɛ in Theorem 22) If η 0 η 3 = 1and ϕ = 1, (1) is equivalent to ψ = 2, which in the present situation is equivalent to g(h 1) 2 = ψ = 2(g 1) + g 2, which, in turn, is equivalent to 4(g 1) = g(h g) But since for any scheme with ϕ = 1, h g, this is not possible If η 0 η 3 = 1and ϕ = 1, (1) is equivalent to = ϕψ and (ϕ 1)(ψ + 1) = 0 ψ = 1and = ϕ is not possible since the three non-standard parameters must have the same sign For any scheme with ϕ = 1 and ψ =, λ = Hence we are considering a scheme with ϕ = ψ = = 1 But such a scheme does not exist as is shown in [9] For comparison with Theorem 32, we have, in the next theorem, replaced g by 2s THEOREM 24 Let (X, R) be a non-symmetric 3-scheme Then H = η 0 D 0 + η 1 D 1 + η 2 D 3 + η 3 D 3 with η i {+1, 1}, η 1 η 2 = 1 is an Hadamard matrix if and only if η 0 η 3 =+1, ϕ = 1 and ψ = In this case ψ = = s for a natural s and H is a skew-checkered Hadamard matrix of order 4s 2 PROOF In view of Lemmas 21 and 23 the only fact we still have to prove is the skewcheckeredness of H If ϕ = 1 (considering, in the terminology of [11], the splitting of a GD-scheme according to case IV), then according to the results of [11, p 34] for a suitable ordering, D 1 D 2 = 0 A 12 A 1h A 21 0 A 2h A h1 A h2 0 where the A ij have entries ±1 and it holds that A ij J = JA ij = 0 and D 3 = (J g I g ) J g As D1 T = D 2, the matrix D 1 D 2 is skew-symmetric Hence H = D 0 + D 1 D 3 + D 3 has the form H = (H ij ), where H ii = J g and H ij J g = J g H ij = 0ifi = j For ψ =, H is an Hadamard matrix and so we see that H is a skew-checkered Hadamard matrix, THEOREM 25 If H is a skew-checkered Hadamard matrix of order 4s 2, then

6 948 R W Goldbach and H L Claasen D 0 = I 4s 2, D 1 = 1 2 (J 4s 2 + H) I 2s J 2s, D 2 = D T 1 = 1 2 (J 4s 2 H), D 3 = I 2s J 2s I 4s 2 are the adjacency matrices of a non-symmetric 3-scheme (X, R) with ϕ = 1 and ψ = = s PROOF Using H 2 = 4s 2 I 4s 2 + 4s(I 2s J 2s ) one finds that the algebra generated by the matrices D 0, D 1, D 2 and D 3 is a commutative four-dimensional algebra, all of whose elements are normal matrices Hence these matrices are the adjacency matrices of a non-symmetric 3-scheme such that D 0 + D 1 D 2 + D 3 is an Hadamard matrix Theorem 24 now implies ϕ = 1and ψ = = s EXAMPLE 26 Then Let x 0 = (+1, +1, 1, 1), x 1 = (+1, 1, +1, 1), x 2 = (+1, 1, 1, +1) J 4 x0 T x 0 x1 T x 1 x2 T x 2 x H = 0 T x 0 J 4 x2 T x 2 x1 T x 1 x 1 T x 1 x2 T x 2 J 4 x0 T x 0 x2 T x 2 x1 T x 1 x0 T x 0 J 4 is a skew-checkered Hadamard matrix of the order 16 (this is a special case of Theorem 41) Hence if D 0 = I 16, D 1 = 1 2 (J 16 + H) I 4 J 4, D 2 = D T 1 = 1 2 (J 16 H), D 3 = I 4 J 4 I 16, then the D i are the adjacency matrices of a non-symmetric 3-scheme (X, R) with ϕ = 1 and ψ = = 2 In a later paper it will be shown that there are exactly two non-symmetric 3-schemes (X, R) connected in the sense of Theorem 24 with a skew-checkered Hadamard matrix of order 16 The non-isomorphic schemes are isospectral, of course, such that ϕ = 1 and ψ = = 2 For one scheme the connected Hadamard matrix is the one given in Example 26, while for the other scheme the Hadamard matrix has eight blocks of rank 1 and eight blocks of rank 2 THEOREM 27 Let (X, R) be a non-symmetric 3-scheme and suppose C = η 1 D 1 +η 2 D 2 + η 3 D 3 with η i { 1, +1} and not all η i equal Then C is not the core of a normalized conference matrix If η 1 η 2 = 1, then C is not a conference matrix PROOF First let η 1 η 2 =+1 (X, R) then, has as its symmetric closure, a pseudo-cyclic 2-scheme But by Theorem 45 in [12] this is not possible (see also [13]) Now let η 1 η 2 = 1 It is easily seen that C cannot be the core of a normalized conference matrix The proof that C is not a conference matrix if η 1 η 2 = 1is left to the reader Its proof is analogous to the one given for Theorem 24 For a non-symmetric 3-scheme (X, R) the non-standard parameters are useful for the proof of other properties as well In [13], among other things, the following properties are demonstrated (1) v = p, p + 1, 2p for any prime p, (2) (X, R) is formally self-dual if and only if ψ =, (3) (X, R) is pseudo-cyclic if and only if ϕ = ψ =

7 Non-symmetric 3-schemes SCHEMES OF TYPE L 1,s (2s) The next theorem is a generalization of a theorem shown in [16] THEOREM 31 Let (X, R) be a symmetric 3-scheme of the type L 1,s (2s) Then for a suitable ordering of the elements of X and of the relations, H = D 0 + D 1 D 2 + D 3 = J 4s 2 2D 2 is a symmetric, checkered Hadamard matrix of order 4s 2 PROOF Let for (X, R), R 0 = R 0, R 1 = R 3, R 2 = R 1 R 2 and Ṽ 0 = V 0, Ṽ 1 = V 3, Ṽ 2 = V 1 V 2 Then the pair (X, R), where R ={ R 0, R 1, R 2 } is a (symmetric) 2-scheme (For a proof of this we refer to [10], but see also [1, 3]) The first eigenvalue matrix P of (X, R) is ( 1 2s 1 ) 2s(2s 1) P = 1 2s 1 2s The scheme is a GD-scheme of type (2s, 2s) As usual the matrices D i are the adjacency matrices of (X, R) and let D k be the ones of (X, R) Now consider D 1 D 2 Since D 1 + D 2 = D 2 is a block matrix with zero blocks on the main diagonal and all other blocks J 2s, one derives easily that D 1 D 2 has zero blocks on the main diagonal and all other blocks are square of order 2s and with entries ±1 Note that (D 1 D 2 )(I 2s J 2s ) = (D 1 D 2 )(D 0 + D 3 ) = 0 From this one easily derives, if D (0) is an off-diagonal block of D 1 D 2, that D (0) J 2s = J 2s D (0) = 0 Now it is easy to complete the proof of the theorem THEOREM 32 Let H be a symmetric, checkered Hadamard matrix of order 4s 2 If D 0 = I 4s 2, D 1 = 1 2 (J 4s 2 + H 2(I 2s J 2s )), D 2 = 1 2 (J 4s 2 H), D 3 = I 2s J 2s I 4s 2, then the matrices D 0,D 1,D 2 and D 3 are the adjacency matrices of a symmetric 3-scheme of type L 1,s (2s) PROOF As in the proof of Theorem 25 one shows that the D i are the adjacency matrices of a symmetric 3-scheme Calculating the intersection numbers from D i D j = pij k D k one sees that the scheme is of type L 1,s (2s) 4 CONSTRUCTIONS OF CHECKERED HADAMARD MATRICES According to Wallis [16] the question of the existence of what we call checkered Hadamard matrices was first raised by K A Bush Several constructions of Hadamard matrices of this type are known (a) In [4], Bush uses complete orthogonal arrays (b) In [16], Wallis uses orthogonal arrays (though not complete ones) and affine designs

8 950 R W Goldbach and H L Claasen (c) In [8], we use cyclotomy in so-called finite 1-rings There we constructed for s = 2 r 1 symmetric 3-schemes of type L 1,s (2s) and non-symmetric 3-schemes of which the symmetric closure is a GD-scheme of type (2s, 2s) Hence for those schemes D 0 + D 1 D 2 + D 3 is a checkered Hadamard matrix For a complete overview we refer to [10] THEOREM 41 Assume that the following hold (1) H = (H ij ) is a square block matrix of order 4s 2 (2) For every i, j {1, 2,,2s}, H ij = x T ij y ji, where x ij and y ij are row vectors of length 2s with entries +1 and 1 with the property that for all i, x ii = y ii =±j, where j is the all-one row vector of length 2s (3) X i and Y i are the square matrices of which the j-th rows are x ij and y ij, respectively Then (1) the rank of H ij is 1 for all i and j; (2) H is a checkered Hadamard matrix if and only if all X i and Y i are Hadamard matrices of order 2s In that case s = 1 or s 0 (mod 2) PROOF Rank(H ij ) = 1 for all i and j is obvious Suppose H is a checkered Hadamard matrix From HH T = 4s 2 I 4s 2 we find δ ij 4s 2 I 2s = u (x T iu y ui)(y T uj x ju) = u (y ui y T uj )(xt iu x ju) Taking i = j we find 2sI 2s = u x T iu x iu = X T i X i Hence the X i are Hadamard matrices From H T H = 4s 2 I 4s 2 it follows that the Y i are Hadamard matrices The converse is proven in a similar way s = 1ors 0 (mod 2) follows from the fact that the X i are Hadamard matrices In Theorem 41 we found all checkered Hadamard matrices under the condition that all blocks H ij have rank 1 Szekeres [15] found several matrices of the form given in Theorem 41 If one allows blocks of higher ranks, then, as we shall show in a later paper, the matter becomes more complicated in that only certain rank distributions are allowed For example, for a checkered Hadamard matrix of order 16 all blocks other than those on the main diagonal cannot have rank 2 From Theorem 41 it directly follows that if there exists an Hadamard matrix of order 2s (s = 1ors even), then there exists a symmetric, checkered Hadamard matrix of order 4s 2 This was first shown by Wallis [16] There the result was reached by a circuitous route Goethals and Seidel [7] showed that if an Hadamard matrix of order 2s exists, then a regular, symmetric Hadamard matrix with constant diagonal of order 4s 2 also exists This is also covered by Theorem 41 In [16] it is noted that there exists a symmetric, checkered Hadamard matrix of order 4s 2 if and only if a strongly regular graph with v = 4s 2, k = s(2s 1), λ = µ = s(s 1) with the following properties exists: the vertices of such a graph can be partitioned into 2s cocliques of size 2s such that a vertex in a given coclique is adjacent to exactly s of the vertices in any of the other cocliques

9 Non-symmetric 3-schemes 951 REFERENCES 1 }E Bannai, Subschemes of some association schemes, J Algebra, 144 (1991), }E Bannai and T Ito, Algebraic Combinatorics I, Association Schemes, Benjamin/Cummings, California, }W G Bridges and R A Mena, On the rational spectra of graphs with Abelian Singer groups, Algebra Applic, 46 (1982), }K A Bush, Unbalanced Hadamard matrices and finite projective planes of even order, J Comb Theory Ser A, 11 (1971), }P Delsarte, An Algebraic Approach to the Association Schemes of Coding Theory, Philips Research Report Supplements, 10 (1973) 6 }H Enomoto and R A Mena, Distance-regular digraphs of girth 4, J Comb Theory Ser B, 43 (1987), }J M Goethals and J J Seidel, Strongly regular graphs derived from combinatorial designs, Can J Math, XXII (1970), }R W Goldbach and H L Claasen, Cyclotomic schemes over finite, commutative, admissible rings, Indagationes Mathematicae, NS, 3 (1992), }R W Goldbach and H L Claasen, On splitting the Clebsch graph, Indagationes Mathematicae, NS, 5 (1994), }R W Goldbach and H L Claasen, Hadamard matrices of a certain block form suggested by K A Bush, Report , Delft University of Technology, Delft, }R W Goldbach and H L Claasen, The structure of imprimitive non-symmetric 3-class association schemes, Eur J Combin, 17 (1996), }R W Goldbach and H L Claasen, Feasibility conditions for non-symmetric 3-class association schemes, Discrete Math, 159 (1996), }R W Goldbach and H L Claasen, The use of non-standard parameters of a non-symmetric 3-class association scheme, to appear as a report of Delft University of Technology 14 }R Mathon, 3-class association schemes, Proc Conf on Algebraic Aspects of Combinatorics, Toronto, 1975, pp }G Szekeres, A new class of symmetric block designs, J Comb Theory, 6 (1969), }W D Wallis, On a problem of K A Bush concerning Hadamard matrices, Bull Austr Math Soc, 6 (1972), }W D Wallis, Ann Penfold Street and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices, Lecture Notes in Mathematics, 292 (1972) Springer, Berlin Received 28 July 1998 and accepted 30 July 1998 R W GOLDBACH Department ITS/TWI, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands RWGoldbach@twitudelftnl