New symmetric designs from regular Hadamard matrices

Yury J. Ionin^{∗} Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859, USA

                                    yury.ionin@cmich.edu

December 19, 1997

Abstract For every positive integer m, we construct a symmetric (v, k, λ)-design with parameters v = \frac{h((2h−1)^{2m}−1)}{h−1} , k = h(2h − 1)^{2m−1}, and λ = h(h − 1)(2h − 1)^{2m−2}, where h = ±3 · 2^{d} and |2h − 1| is a prime power. For m ≥ 2 and d ≥ 1, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order 9 · 4^{d}.

Submitted: October 30, 1997; Accepted: November 17, 1997

MR Subject Number: 05B05

Keywords: Symmetric design, regular Hadamard matrix, balanced generalized weighing matrix

1 Introduction

Let v > k > λ ≥ 0 be integers. A symmetric (v, k, λ)-design is an incidence structure (P, B), where P is a set of cardinality v (the point-set) and B is a family of v k-subsets (blocks) of P such that any two distinct points are contained in exactly λ blocks. If P = {p_{1}, ..., p_{v}} and B = {B_{1}, ..., B_{v}}, then the (0, 1)-matrix M = [m_{ij}] of order v, where m_{ij} = 1 if and only if p_{j} ∈ B_{i}, is the incidence matrix of the design. A (0, 1)-matrix X of order v is the incidence matrix of a symmetric (v, k, λ)-design if and only if it satisfies the equation XX^{T} = (k − λ)I + λJ, where I is the identity matrix and J is the all-one matrix of order v. For references, see [1] or [3, Chapter 5].

A Hadamard matrix of order n is an n by n matrix H with entries equal to ±1 satisfying HH^{T} = nI. A Hadamard matrix is regular if its row and column sums are constant. This sum is always even and if we denote it 2h, then the order of the matrix is equal to 4h^{2}. Replacing −1s in a regular Hadamard matrix of order 4h^{2} by 0s yields the incidence matrix of a symmetric (4h^{2}, 2h^{2} − h, h^{2} − h)-design usually called a Menon design. Conversely, replacing 0s by −1s in the incidence matrix of a symmetric (4h^{2}, 2h^{2} − h, h^{2} − h)-design yields a regular Hadamard matrix of order 4h^{2}. For references, see [9]. In this paper, we will be interested in regular Hadamard matrices of order 9 · 4^{d}, where d is a positive integer. If H is such a matrix, then the Kronecker product of a regular Hadamard matrix of order 4 and H is a regular Hadamard matrix of order 9 · 4^{d+1}. Therefore, one can obtain a family of regular Hadamard matrices of order 9 · 4^{d}, starting with a regular Hadamard matrix of order 36.

∗The author acknowledges with thanks the Central Michigan University Research Professor award.

A balanced generalized weighing matrix BGW(v, k, λ) over a (multiplicatively written) group G is a matrix W = [ω_{ij}] of order v with entries from the set G ∪ {0} such that (i) each row and each column of W contain exactly k non-zero entries and (ii) for any distinct rows i and h, the multiset

{ω_{hj}^{−1}ω_{ij}: 1 ≤ j ≤ v, ω_{ij} ≠ 0, ω_{hj} ≠ 0}

contains exactly λ/|G| copies of every element of G.

In this paper, we will use a balanced generalized weighing matrix BGW(q^{m} + q^{m−1} + ⋯ + q + 1, q^{m}, q^{m} − q^{m−1}) over a cyclic group G of order t, where q is a prime power, m is a positive integer, and t is a divisor of q − 1. Such matrices are known to exist [3, IV.4.22] and have been applied to constructing symmetric designs by Rajkundlia [8], Brouwer [2], Fanning [4], and the author [5, 6]. If M is a set of m by n matrices, G is a group of bijections M → M, and W is a balanced generalized weighing matrix over G, then, for any P ∈ M, W ⊗ P denotes the matrix obtained by replacing every entry σ in W by the matrix σP. In Section 2 (Lemma 2.1), we will prove the following modification of a result from [6]:

Let M be a set of matrices of order v containing the incidence matrix M of a symmetric (v, k, λ)-design with q = \frac{k^{2}}{k−λ} a prime power. Let G be a finite cyclic group of bijections M → M such that (i) (σP)(σQ)^{T} = P Q^{T} for any P, Q ∈ M and σ ∈ G, (ii) ∑_{σ∈G} σM = \frac{k|G|}{v}J, and (iii) |G| divides q − 1. If W is a balanced generalized weighing matrix BGW(q^{m} + ⋯ + q + 1, q^{m}, q^{m} − q^{m−1}) over G, then W ⊗ M is the incidence matrix of a symmetric (v(q^{m} + q^{m−1} + ⋯ + q + 1), kq^{m}, λq^{m})-design.

In order to apply this lemma, we need a symmetric (v, k, λ)-design to start with. In the paper [6], we have shown that the designs corresponding to certain McFarland and Spence difference sets (or their complements) serve as such starters. In Section 3 of this paper, we show that for h = ±3 · 2^{d}, if |2h − 1| is a prime power, then there is a symmetric (4h^{2}, 2h^{2} − h, h^{2} − h)-design, which can also serve as a starter. As a result, we show that for any positive integers m and d, if h = ±3 · 2^{d} and |2h − 1| is a prime power, then there exists a symmetric (v, k, λ)-design with

v = \frac{h((2h − 1)^{2m} − 1)}{h − 1}, k = h(2h − 1)^{2m−1}, λ = h(h − 1)(2h − 1)^{2m−2}.

These parameters are new, except m = 1 (Menon designs) and d = 0 (constructed by the author in [6]).

2 Preliminaries

Throughout this paper, we will denote identity, zero, and all-one matrices of suitable orders by I, O, and J, respectively.

If W is a balanced generalized weighing matrix of order w over a group G of bijections on a set M of matrices of order n, then, for any P ∈ M, we will denote by W ⊗ P the matrix of order nw obtained by replacing every nonzero entry σ in W by the matrix σP and every zero entry in W by the zero matrix of order n.

The following lemma represents a slight modification of a result proven in [6]. Since it is crucial for this paper and the proof is short, we will repeat it here.

Lemma 2.1 Let v > k > λ ≥ 0 be integers. Let M be a set of matrices of order v and G a finite group of bijections M → M satisfying the following conditions:

(i) M contains the incidence matrix M of a symmetric (v, k, λ)-design; (ii) for any P, Q ∈ M and σ ∈ G,

(σP)(σQ)^{T} = P Q^{T};

(iii) ∑_{σ∈G} σM = \frac{k|G|}{v}J;

(iv) q = \frac{k^{2}}{k−λ} is a prime power;

(v) G is cyclic and |G| divides q − 1.

Then, for any positive integer m, there exists a symmetric (vw, kq^{m}, λq^{m})-design, where w = \frac{q^{m+1}−1}{q−1}.

Proof. Let W = [ω_{ij}], i, j = 1, 2, …, w be a balanced generalized weighing matrix BGW(w, q^{m}, q^{m} − q^{m−1}) over G. We claim that W ⊗ M is the incidence matrix of a symmetric (vw, kq^{m}, λq^{m})-design. It suffices to show that, for i, h = 1, 2, …, w,

∑_{j=1}^{w}(ω_{ij}M)(ω_{hj}M)^{T} = \begin{cases}(k − λ)q^{m}I + λq^{m}J if i = h, \\ λq^{m}J if i ≠ h. \end{cases}

If i = h, we have for some σ_{j} ∈ G,

∑_{j=1}^{w}(ω_{ij}M)(ω_{hj}M)^{T} = ∑_{j=1}^{q^{m}}(σ_{j}M)(σ_{j}M)^{T} = ∑_{j=1}^{q^{m}} MM^{T} = (k − λ)q^{m}I + λq^{m}J.

If i ≠ h, we have for some σ_{j}, τ_{j} ∈ G,

∑_{j=1}^{w}(ω_{ij}M)(ω_{hj}M)^{T} = ∑_{j=1}^{q^{m}−q^{m−1}}(σ_{j}M)(τ_{j}M)^{T} = ∑_{j=1}^{q^{m}−q^{m−1}}(τ_{j}^{−1}σ_{j}M)M^{T}

= \frac{q^{m} − q^{m−1}}{|G|} ( ∑_{σ∈G} σM)M^{T} = \frac{k(q^{m} − q^{m−1})}{v}JM^{T} = \frac{k^{2}(q^{m} − q^{m−1})}{v}J = λq^{m}J.

□ Definition 2.2 Let v > k > λ > 0 be integers. A (v, k, λ)-difference set is a k-subset of an (additively written) group Γ of order v such that the multiset {x − y: x, y ∈ Γ} contains exactly λ copies of each nonzero element of Γ.

Several infinite families of difference sets are known (see [3] or [7] for references). We will mention the McFarland family having parameters (p^{d+1}(r+1), p^{d}r, p^{d−1}(r−1)), where p is a prime power, d is a positive integer, and r = \frac{p^{d+1}−1}{p−1}, and the Spence family having parameters (3^{d+1}(3^{d+1}−1)/2, 3^{d}(3^{d+1}+1)/2, 3^{d}(3^{d}+1)/2), where d is a positive integer.

If Δ is a (v, k, λ)-difference set in a group Γ and B = {Δ + x: x ∈ Γ}, then dev(Δ) = (Γ, B) is a symmetric (v, k, λ)-design.

In order to apply Lemma 2.1, we need a symmetric (v, k, λ)-design with q = \frac{k^{2}}{k−λ} a prime power, a set M of matrices of order v containing the incidence matrix this design, and a cyclic group G satisfying conditions (ii), (iii), and (v) of Lemma 2.1. In the paper [6], we have shown that (v, k, λ) can be the parameters of any McFarland or Spence difference set or their complement with q = \frac{k^{2}}{k−λ} a prime power. In this paper, we will use the Spence (36, 15, 6)-difference set in Γ = ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{4} and the complementary (36, 21, 12)-difference set. In the next section, we will reproduce the construction of the corresponding M and G given in [6]

3 (36, 15, 6)- and (36, 21, 12)-difference sets

We start with a brief description of the Spence (36, 15, 6)-difference set in Γ = ℤ_{3} ⊕ ℤ_{3} ⊕ ℤ_{4}.

We consider Γ as the set of triples (x_{1}, x_{2}, x_{3}), where x_{1}, x_{2} ∈ {0, 1, 2} and x_{3} ∈ {0, 1, 2, 3} with the mod 3 and the mod 4 addition, respectively. Consider ℤ_{3} ⊕ ℤ_{3} as a 2-dimensional vector space over the field GF(3). Let L_{1}, L_{2}, L_{3}, L_{4} be its 1dimensional subspaces. Put D_{1} = {(x_{1}, x_{2}, 0) ∈ Γ: (x_{1}, x_{2}) ∉ L_{1}} and, for i = 2, 3, 4, D_{i} = {(x_{1}, x_{2}, i − 1) ∈ Γ: (x_{1}, x_{2}) ∈ L_{i}}. Then D = D_{1} ∪ D_{2} ∪ D_{3} ∪ D_{4} is a (36, 15, 6)- difference set in Γ [7, Theorem 11.2].

In order to obtain the incidence matrix of the corresponding symmetric design, we have to select an order on Γ. We will assume that (x_{1}, x_{2}, x_{3}) precedes (y_{1}, y_{2}, y_{3}) in Γ if and only if there is i such that x_{i} < y_{i} and x_{j} = y_{j} whenever j > i. Let M be the (0, 1)-matrix of order 36 whose rows and columns are indexed by elements of Γ in this order and (x, y)-entry is equal to 1 if and only if y − x ∈ D. Then M is the incidence matrix of a symmetric (36, 15, 6)-design. In order to describe the structural properties of M which will be important in the sequel, we introduce the following operation ρ on the set of 3 by 3 block-matrices.

Definition 3.1 Let P = [P_{ij}] be a 3 by 3 block-matrix with square blocks (in particular, P can be a 3 by 3 matrix). Denote by ρP the matrix obtained by applying the cyclic permutation ρ = (123) of degree 3 to the set of columns of P, i.e.,

ρ [\begin{matrix}P_{11} & P_{12} & P_{13} \\ P_{21} & P_{22} & P_{23} \\ P_{31} & P_{32} & P_{33} \end{matrix}] = [\begin{matrix}P_{13} & P_{11} & P_{12} \\ P_{23} & P_{21} & P_{22} \\ P_{33} & P_{31} & P_{32} \end{matrix}] .

The above incidence matrix M of a symmetric (36, 15, 6)-design can be represented as a 4 by 4 block-matrix

M = [\begin{matrix}M_{1} & M_{2} & M_{3} & M_{4} \\ M_{4} & M_{1} & M_{2} & M_{3} \\ M_{3} & M_{4} & M_{1} & M_{2} \\ M_{2} & M_{3} & M_{4} & M_{1} \end{matrix}] ,

where each M_{i} is a 9 by 9 matrix. Further, each M_{i} can be represented as a 3 by 3 block-matrix

M_{i} = [\begin{matrix}M_{i1} & M_{i2} & M_{i3} \\ M_{i3} & M_{i1} & M_{i2} \\ M_{i2} & M_{i3} & M_{i1} \end{matrix}] ,

where each M_{ij} is a matrix of order 3, M_{11} = O, M_{12} = M_{13} = J, M_{21} = M_{22} = M_{23} = M_{31} = M_{41} = I, M_{32} = M_{43} = ρI, and M_{33} = M_{42} = ρ^{2}I.

Let M be the set of block-matrices P = [P_{ij}], i, j = 1, 2, 3, 4, where each P_{ij} is a block-matrix P_{ij} = [P_{ijkl}], k, l = 1, 2, 3, satisfying the following conditions:

(i) each P_{ijkl} is a (0, 1)-matrix of order 3;

(ii) for i = 1, 2, 3, 4, there is a unique h_{i} = h_{i}(P) ∈ {1, 2, 3, 4} such that

(P_{ijk1}, P_{ijk2}, P_{ijk3}) is a permutation of (O, J, J) for j = h_{i} and all k

and

P_{ijkl} ∈ {I, ρI, ρ^{2}I} for j ≠ h_{i} and all k, l.

Clearly, the above matrix M is an element of M.

Define a bijection σ: M → M by σP = P′, where

(i) for i = 1, 2, 3, 4 and j = 2, 3, 4, P′_{ij} = P_{i,j−1};

(ii) for i = 1, 2, 3, 4, if h_{i} = 4, then P′_{i1} = ρP_{i4};

(iii) for i = 1, 2, 3, 4, if h_{i} ≠ 4, then P′_{i1kl} = ρP_{i4kl} for all k, l.

Let G be the cyclic group generated by σ. Then |G| = 12.

Claim. For any P, Q ∈ M, (σP)(σQ)^{T} = P Q^{T}. Proof. Let P, Q ∈ M and let P′ = σP and Q′ = σQ. It suffices to show that, for i = 1, 2, 3, 4,

P′_{i1}Q_{i1}^{′T} = P_{i4}Q_{i4}^{T}.  (1)

If h_{i}(P) = h_{i}(Q) = 4 or h_{i}(P) ≠ 4 and h_{i}(Q) ≠ 4, then P′_{i1} is obtained from P_{i4} by the same permutation of columns as Q′_{i1} from Q_{i4}, so (1) is clear. Suppose h_{i}(P) = 4 and h_{i}(Q) ≠ 4. Then (P_{i4k1}, P_{i4k2}, P_{i4k3}) is a permutation of (O, J, J) and matrices Q_{i4k1}, Q_{i4k2}, Q_{i4k3} have the same row sum (equal to 1). Therefore

∑_{l=1}^{3} P′_{i1kl}Q_{i1kl}^{′T} = ∑_{l=1}^{3} P_{i4kl}Q_{i4kl}^{T} = 2J,

and (1) follows. □

It is readily verified that

∑_{n=0}^{11} σ^{n}M = 5J.  (2)

Thus, the set M, the matrix M, and the group G satisfy Lemma 2.1 for (v, k, λ) = (36, 15, 6) with |G| = 12. Note that the sum of the entries of any row of any matrix P ∈ M is equal to 15.

Let \overline{M} = J −M and \overline{M} = \{J −P: P ∈ M\}. Without changing G, we obtain that \overline{M}, \overline{M}, and G satisfy Lemma 2.1 for (v, k, λ) = (36, 21, 12). The sum of the entries of any row of any matrix P ∈ \overline{M} is equal to 21.

Note that the described (36, 15, 6)-design and (36, 21, 12)-design are symmetric (4h^{2}, 2h^{2} − h, h^{2} − h)-designs with h = 3 and h = −3, respectively.

4 Using the Kronecker product

The next lemma will allow us to double the parameter h in a family of symmetric (4h^{2}, 2h^{2} − h, h^{2} − h)-designs satisfying Lemma 2.1.

Lemma 4.1 Let an integer h ≠ 0, a set M of matrices of order 4h^{2}, and a finite cyclic group G = ⟨σ⟩ of bijections M → M satisfy the following conditions:

(i) M contains the incidence matrix M of a symmetric (4h^{2}, 2h^{2}−h, h^{2}−h)-design; (ii) for any P, Q ∈ M, (σP)(σQ)^{T} = P Q^{T}; (iii) ∑_{n=0}^{|G|−1} σ^{n}M = \frac{(2h−1)|G|}{4h}J. (iv) the sum of the entries of any row of any matrix P ∈ M is equal to 2h^{2} − h. Then there exists a set M_{1} of matrices of order 16h^{2} and a cyclic group G_{1} = ⟨τ⟩

of bijections M_{1} → M_{1} satisfying the following conditions:

(a) M_{1} contains the incidence matrix M_{1} of a symmetric (16h^{2}, 8h^{2}−2h, 4h^{2}−2h)- design;

(b) for any R, S ∈ M_{1}, (τR)(τS)^{T} = RS^{T};

(c) ∑_{n=0}^{|G_{1}|−1} τ^{n}M_{1} = \frac{(4h−1)|G_{1}|}{8h}J;

(d) the sum of the entries of any row of any matrix R ∈ M_{1} is equal to 8h^{2} − 2h;

(e) |G_{1}| = 2|G|.

Proof. For any P ∈ M, define

R_{P} = [\begin{matrix}J − P & P & P & P \\ P & J − P & P & P \\ P & P & J − P & P \\ P & P & P & J − P \end{matrix} ] .

It is well known and readily verified that M_{1} = R_{M} is the incidence matrix of a symmetric (16h^{2}, 8h^{2} − 2h, 4h^{2} − 2h)-design.

Let M_{1} = {R_{P} : P ∈ M}. Then M_{1} ∈ M_{1}, so M_{1} satisfies (a). Condition (d) is implied by (iv). Any matrix R ∈ M_{1} can be divided into eight 4h^{2} by 8h^{2} cells R_{ij}, 1 ≤ i ≤ 4, 1 ≤ j ≤ 2. Observe that each R_{ij} is of one of the two following types:

(type 1) R_{ij} = [P J − P] or R_{ij} = [J − P P], P ∈ M;

(type 2) R_{ij} = [P P], P ∈ M.

Observe also that R_{i1} and R_{i2} are not of the same type.

For any R ∈ M_{1}, denote by τR a (0, 1)-matrix of order 16h^{2} divided into eight 4h^{2} by 8h^{2} cells τR_{ij}, 1 ≤ i ≤ 4, 1 ≤ j ≤ 2, where

τR_{i2} = R_{i1}

and

τR_{i1} = \begin{cases}J − R_{i2} & if R_{i2} is of type 1, \\ [σP σP] & if R_{i2} = [P P] . \end{cases}

In order to verify (b), it suffices to show that, for i = 1, 2, 3, 4, (τR_{i1})(τS_{i1})^{T} = R_{i2}S_{i2}^{T}.

If R_{i2} and S_{i2} are of type (1), then (τR_{i1})(τS_{i1})^{T} = (J − R_{i2})(J − S_{i2})^{T} = 8h^{2}J − R_{i2}J^{T} − JS_{i2}^{T} + R_{i2}S_{i2}^{T} = R_{i2}S_{i2}^{T} for the row sum of any matrix of type 1 is equal to 4h^{2}. If R_{i2} = [P P] and S_{i2} = [Q Q], where P, Q ∈ M, then (τR_{i1})(τS_{i1})^{T} = 2(σP)(σQ)^{T} = 2P Q^{T} = R_{i2}S_{i2}^{T}. If R_{i2} = [P P] and S_{i2} is of type 1, then (τR_{i1})(τS_{i1})^{T} = (σP)J = (2h^{2} − h)J = R_{i2}S_{i2}^{T}.

Let G_{1} be the group of bijections M_{1} → M_{1} generated by τ. Then (e) is satisfied, and we have to verify (c). For n = 1, 2, …, 2|G| − 1, let A_{n} be the (i, j)-block of the 4 by 4 block-matrix τ^{n}M_{1}. Then there is P ∈ M such that the multiset {A_{n}: 0 ≤ n ≤ 2|G| − 1} is the union of {σ^{n}P: 0 ≤ n ≤ |G| − 1} and the multiset consisting of \frac{|G|}{2} copies of P and \frac{|G|}{2} copies of J − P. Therefore,

∑_{n=0}^{2|G|−1} A_{n} = ∑_{n=0}^{|G|−1} σ^{n}P + \frac{|G|}{2}J = \frac{(2h − 1)|G|}{4h}J + \frac{|G|}{2}J = \frac{(4h − 1)|G_{1}|}{8h}J.

□ The following theorem is now immediate by induction.

Theorem 4.2 Let an integer h ≠ 0, a set M of matrices of order 4h^{2}, and a finite cyclic group G of bijections M → M satisfy conditions (i)–(iv) of Lemma 4.1. Then, for any positive integer d, there exists a non-empty set M_{d} of matrices of order 4^{d+1}h^{2} and a cyclic group G_{d} of bijections M_{d} → M_{d} satisfying the following conditions:

(a) M_{d} contains the incidence matrix M_{d} of a symmetric design with parameters

(4^{d+1}h^{2}, 2^{2d+1}h^{2} − 2^{d}h, 2^{2d}h^{2} − 2^{d}h);

(b) for any P, Q ∈ M_{d} and τ ∈ G_{d}, (τP)(τQ)^{T} = P Q^{T}; (c) ∑_{τ∈G_{d}} τM_{d} = \frac{(2^{d+1}h−1)|G_{d}|}{2^{d+2}h}J;

(d) the sum of the entries of any row of any matrix R ∈ M_{d} is equal to 2^{2d+1}h^{2} − 2^{d}h;

(e) |G_{d}| = 2^{d}|G|.

We combine Theorem 4.2 and Lemma 2.1 and obtain the main result of this paper.

Theorem 4.3 If h = ±3 · 2^{d}, where d is a positive integer and |2h − 1| is a prime power, then, for any positive integer m, there exists a symmetric (\frac{h((2h−1)^{2m}−1)}{h−1}, h(2h− 1)^{2m−1}, h(h − 1)(2h − 1)^{2m−2})-design.

Proof. We start with the set M or \overline{M} described in Section 3 and apply Theorem 4.2 to this set to obtain the set of matrices M_{d} or \overline{M}_{d} and the group G_{d}. Then we apply Lemma 2.1. Properties (ii) and (iii) required in Lemma 2.1 are implied by (b) and (c) of Theorem 4.2. The parameter q of Lemma 2.1 is equal to (2h_{d} − 1)^{2}, where h_{d} = ±3 · 2^{d}, so q is a prime power. Since |G| = 12, we have |G_{d}| = 3 · 2^{d+2} = 4|h_{d}|, so |G_{d}| divides q − 1. □

Remark 4.4 These parameters are new, except m = 1 (Menon designs).

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