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 or is a prime power, then there exists a biregular Hadamard matrix of order 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.
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DOI: 10.5802/alco.33
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}, pages = {697--722}, publisher = {MathOA foundation}, volume = {1}, number = {5}, year = {2018}, doi = {10.5802/alco.33}, zbl = {1401.05054}, mrnumber = {3887408}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.33/} }
TY - JOUR AU - Hirasaka, Mitsugu AU - Momihara, Koji AU - Suda, Sho TI - A new approach to the excess problem of Hadamard matrices JO - Algebraic Combinatorics PY - 2018 SP - 697 EP - 722 VL - 1 IS - 5 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.33/ DO - 10.5802/alco.33 LA - en ID - ALCO_2018__1_5_697_0 ER -
%0 Journal Article %A Hirasaka, Mitsugu %A Momihara, Koji %A Suda, Sho %T A new approach to the excess problem of Hadamard matrices %J Algebraic Combinatorics %D 2018 %P 697-722 %V 1 %N 5 %I MathOA foundation %U http://archive.numdam.org/articles/10.5802/alco.33/ %R 10.5802/alco.33 %G en %F ALCO_2018__1_5_697_0
Hirasaka, Mitsugu; Momihara, Koji; Suda, Sho. 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. http://archive.numdam.org/articles/10.5802/alco.33/
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