Implementing Hadamard Matrices in SageMath

TL;DR

This paper presents an implementation of Hadamard and skew Hadamard matrices in SageMath, enabling verification and construction of these matrices for all known orders up to 1000, including additional related mathematical objects.

Contribution

The authors developed SageMath implementations for constructing and verifying Hadamard and skew Hadamard matrices up to order 1000, including new tools for related objects like difference sets.

Findings

Successfully constructed all known Hadamard matrices up to order 1000.

Identified and corrected an error in the claimed construction for order 292.

Provided tools for verifying existence and construction of Hadamard matrices.

Abstract

Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.