Hadamard Matrices Modulo p and Small Modular Hadamard Matrices

Abstract

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p-modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p-modular version of the Hadamard conjecture for all but finitely many cases when p ≡ 3 (mod 4), and prove a weaker result for p ≡ 1 (mod 4). Finally, we look at constraints on the existence of m-modular Hadamard matrices when the size of the matrix is small compared to m.A few basic results, all presented in [5], allow us to completely decide the existence of Hadamard matrices modulo 2, 3, 4, 6, 8, and 12.Lemma 2.1.[5] Assume n ≥ 3, and let H be an MH(n, m). Then (m, 4)|n. If n is odd and n ≡ 0 (mod m), then let 1 ≤ r ≤ m − 1 such that r ≡ 2 φ(m)−2 n (mod m); we know that n ≥ 4r.Lemma 2.2. [5] Let H be an MH(n, m), with (n, m) = 1 and n odd. Then n is a quadratic residue of m.Proof. Since H H ≡ nI (mod m), (det H ) 2 ≡ n n (mod m), so n can not be both odd and a nonresidue of m. Lemma 2.3. [5]If n ≡ 0 (mod m) or n ≡ 4 (mod m), then there exists an MH(n, m).Proof. If n ≡ 0 (mod m), then the matrix J is an MH(n, m). If n ≡ 4 (mod m), then the matrix J − 2I is an MH(n, m). Lemma 2.4. [5]Let H 1 be an MH(n 1 , m 1 ) and H 2 be an MH(n 2 , m 2 ). Then H 1 ⊗ H 2 is an MH(n 1 n 2 , (m 1 m 2 , n 1 m 2 , n 2 m 1 )).The operation, described in Lemma 2.4 is most commonly done when one of the components is the real Hadamard matrixin which case it is called "doubling." Theorem 2.5. If n ≥ 3, then (a) an MH(n, 2) exists ⇐⇒ n is even. (b) an MH(n, 3) exists ⇐⇒ n ≡ 5 (mod 6). (c) an MH(n, 4) exists ⇐⇒ n ≡ 0 (mod 4). (d) an MH(n, 6) exists ⇐⇒ n is even. (e) an MH(n, 8) exists ⇐⇒ n ≡ 0 (mod 4). (f) an MH(n, 12) exists ⇐⇒ n ≡ 0 (mod 4). Journal of Combinatorial Designs DOI 10.1002/jcd P-MODULAR HADAMARD MATRICES 395 Proof. Cases (a) through (d) are addressed in [5]. An MH(n, 8) can only exist by Lemma 2.1 if n ≡ 0, 4 (mod 8) and both of these are constructed in Lemma 2.3. By Lemma 2.1, an MH(n, 12) can only exist if n ≡ 0, 4, 8 (mod 12). The n ≡ 0, 4, (mod 12) cases are constructed in Lemma 2.3; for the n ≡ 8 (mod 12) case, we can double an MH(6k + 4, 6) for any k ≥ 0 to get an MH(12k + 8, 12).The study of modular symmetric designs was explored in [5] to address the question of deciding the existence of MH(n, m) matrices for m = 5. MODULAR SYMMETRIC DESIGNSIn much the same way that modular Hadamard matrices are a generalization of Hadamard matrices, we can generalize modular symmetric designs from symmetric designs, leading to the following definition.and that DJ ≡ J D ≡ kJ for some integers k, λ, and where J is the v × v matrix of all 1's. Such a design is denoted by its parameters (v, k, λ; m).We may go between modular symmetric designs and modular Hadamard matrices by replacing the −1 en...