Hadamard matrices modulo p and small modular Hadamard matrices

TL;DR

This paper investigates the existence of Hadamard matrices modulo primes using modular symmetric designs, solving specific cases of the modular Hadamard conjecture and proposing a general conjecture for their existence.

Contribution

It introduces a new approach using modular symmetric designs, solves the 7- and 11-modular Hadamard conjecture cases, and formulates a conjecture for general prime moduli.

Findings

Solved 7- and 11-modular Hadamard conjecture cases for all but finitely many instances.

Proposed a conjecture for the existence of p-modular Hadamard matrices for all but finitely many primes.

Established conditional results when 2 is a primitive root of p, especially for p ≡ 3 mod 4.

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 conjecture for a 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 $p \equiv 3 \pmod{4}$, and prove a weaker result for $p \equiv 1 \pmod{4}$. Finally, we look at constraints on the existence of $m$-modular Hadamard matrices when the size of the matrix is small compared to $m$.