A database of constructions of Hadamard matrices

TL;DR

This paper provides a comprehensive database of Hadamard and skew Hadamard matrices up to order 1208, verifies existing results, corrects a known construction, and updates tables for orders up to 2999, enhancing the understanding of their existence and properties.

Contribution

It offers an extensive, verified database of Hadamard matrices, corrects a specific skew Hadamard matrix construction, and updates key tables relating matrix orders to exponents, with insights into Riesel numbers and Paley constructions.

Findings

Verified all known Hadamard matrices up to order 1208.

Corrected a skew Hadamard matrix construction at order 292.

Updated tables for matrix orders up to 2999, improving over 100 entries.

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 $m$.