We use modular symmetric designs to study the existence of Hadamard matricesmodulo certain primes. We solve the $7$-modular and $11$-modular versions ofthe Hadamard conjecture for all but a finite number of cases. In doing so, westate a conjecture for a sufficient condition for the existence of a$p$-modular Hadamard matrix for all but finitely many cases. When $2$ is aprimitive root of a prime $p$, we conditionally solve this conjecture andtherefore the $p$-modular version of the Hadamard conjecture for all butfinitely 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$m$.
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机译：我们使用模块化对称设计来研究对某些素数进行模数调节的Hadamard矩阵的存在。我们为有限数量的情况解决了Hadamard猜想的$ 7 $模数和$ 11 $模数版本。这样做时，我们提出了一个猜想，对于除了有限数量的所有情况以外的所有情况，存在一个满足p $$模数的Hadamard矩阵的充分条件。当$ 2 $是质数$ p $的素数根时，我们有条件地解决这个猜想，因此，在$ p \ equiv 3 \ pmod {4} $的所有无数情况下，Hadamard猜想的$ p $模块化版本，并证明$ p \ equiv 1 \ pmod {4} $的结果较弱。最后，当矩阵的大小比$ m $小时，我们看一下对$ m $模Hadamard矩阵存在的约束。
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