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