An update on modular Hadamard matrices

For \(m \in \mathbb {N}\) $m\in \mathbb{N}$ , an m-modular Hadamard matrix is a square matrix of order n with coefficients in \(\{\pm 1\}\) $\{\pm 1\}$ satisfying \( H H^\top \equiv nI_n \bmod m. \) $$H{H}^{\top }\equiv n{I}_{n}modm.$$ This notion was introduced by Marrero and Butson in the early 1970’s. The case \(m=0\) $m=0$ corresponds to true Hadamard matrices, with rows pairwise orthogonal over \(\mathbb {Z}\) $\mathbb{Z}$ rather than \(\mathbb {Z}/m\mathbb {Z}\) $\mathbb{Z}/m\mathbb{Z}$ . The m-modular Hadamard conjecture, a consequence of its classical counterpart from 1893, postulates the existence of m-modular Hadamard matrices of any order \(n \equiv 0 \bmod 4\) $n\equiv 0mod4$ . It is currently solved for \(m=5, 12, 32\) $m=5,12,32$ and their divisors only. In this paper, we make some partial progress toward the open case \(m=64\) $m=64$ , by constructing 64-modular Hadamard matrices of all orders \(n=4\ell \) $n=4\ell$ such that \(\ell \equiv 3 \bmod 16\) $\ell \equiv 3mod16$ or \(\ell \equiv 7 \bmod 32\) $\ell \equiv 7mod32$ . This includes \(n=668\) $n=668$ and 716, the smallest two of the three open cases \(n \le 1000\) $n\le 1000$ in Hadamard’s conjecture proper, leaving \(n=892\) $n=892$ as the smallest open case in the 64-modular version. Our solutions rest on the construction of relevant families of 64-modular Golay quadruples and the Goethals–Seidel array.