New families of Hadamard matrices with maximum excess

TL;DR

This paper introduces new families of Hadamard matrices with maximum excess, constructed by modifying known matrices derived from quadratic residues, and establishes conditions for their regularity and biregularity.

Contribution

It presents methods to generate regular and biregular Hadamard matrices with maximum excess using row and column negations of quadratic residue-based matrices, and provides conditions for their regularity.

Findings

Existence of biregular Hadamard matrices when certain prime powers conditions are met.

A sufficient condition for quadratic residue Hadamard matrices to be regular.

Construction of Hadamard matrices with maximum excess for specific orders.

Abstract

In this paper, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. In particular, we show that if either $4m^2+4m+3$ or $2m^2+2m+1$ is a prime power, then there exists a biregular Hadamard matrix of order $n=4(m^2+m+1)$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to be regular in terms of four-class translation association schemes on finite fields.