Hadamard Matrices Modulo p and Small Modular Hadamard Matrices

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 , and prove a weaker result for . Finally, we look at constraints on the existence of m-modular Hadamard matrices when the size of the matrix is small compared to m.