A database of constructions of Hadamard matrices

Abstract

Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and skew Hadamard matrices in the open-source software SageMath. This allowed us to verify the correctness of results given in the literature. Within this range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix. We also produce the up to date tables, for $n \le 2999$ (resp. $n\le 999$ for skew case), of the minimum exponents $m$ such that a (skew) Hadamard matrix of order $2^m n$ is known, improving over 100 entries in the previously published sources. We explain how tables' entries are related to Riesel numbers. As a by-product of the latter, we show that the Paley constructions of (skew-)Hadamard matrices do not work for the order $2^m 509203$, for any $m$.