# ALGEBRAIC COMBINATORICS

A new approach to the excess problem of Hadamard matrices
Algebraic Combinatorics, Volume 1 (2018) no. 5, p. 697-722
In this paper, we give a new technique to find families of Hadamard matrices with maximum excess. In particular, 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. More precisely, we show that if either ${\left(2m+1\right)}^{2}+2$ or ${m}^{2}+{\left(m+1\right)}^{2}$ is a prime power, then there exists a biregular Hadamard matrix of order $n={\left(2m+1\right)}^{2}+3$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to regular ones in terms of four-class translation association schemes on finite fields. The core part of this paper is how to find “switching” sets of rows and columns.
Revised : 2018-07-10
Accepted : 2018-07-10
Published online : 2018-11-30
DOI : https://doi.org/10.5802/alco.33
Classification:  05B20,  05B05,  05E30,  11T22,  11T24
Keywords: Hadamard matrix, Regular Hadamard matrix, Biregular Hadamard matrix, Excess, Association scheme, $t$-intersection set, Block design
@article{ALCO_2018__1_5_697_0,
author = {Hirasaka, Mitsugu and Momihara, Koji and Suda, Sho},
title = {A new approach to the excess problem of Hadamard matrices},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {5},
year = {2018},
pages = {697-722},
doi = {10.5802/alco.33},
zbl = {1401.05054},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_5_697_0}
}

A new approach to the excess problem of Hadamard matrices. Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 697-722. doi : 10.5802/alco.33. https://alco.centre-mersenne.org/item/ALCO_2018__1_5_697_0/

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