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A new approach to the excess problem of Hadamard matrices. (English) Zbl 1401.05054
Summary: 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 \((2m+1)^2+2\) or \(m^2+(m+1)^2\) is a prime power, then there exists a biregular Hadamard matrix of order \(n=(2m+1)^2+3\) 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.

MSC:
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B05 Combinatorial aspects of block designs
05E30 Association schemes, strongly regular graphs
11T22 Cyclotomy
11T24 Other character sums and Gauss sums
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[1] Bannai, Eiichi, Subschemes of some association schemes, J. Algebra, 144, 1, 167-188, (1991) · Zbl 0762.20004
[2] Bannai, Eiichi; Ito, Tatsuro, Algebraic Combinatorics I. Association Schemes, xxiv+425 p. pp., (1984), Benjamin/Cumming Publishing Company · Zbl 0555.05019
[3] Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S., Gauss and Jacobi Sums, xi+583 p. pp., (1997), Wiley · Zbl 0906.11001
[4] Best, Marc R., The excess of Hadamard matrix, Indagationes Math., 80, 357-361, (1977) · Zbl 0366.05016
[5] Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried, Design Theory, vol. 1, 2, 69, 78, (1999), Cambridge University Press · Zbl 0945.05005
[6] Brouwer, Andries E.; Wilson, Richard M.; Xiang, Qing, Cyclotomy and strongly regular graphs, J. Algebr. Comb., 10, 1, 25-28, (1999) · Zbl 0929.05094
[7] Craigen, Robert; Kharaghani, Hadi, Weaving Hadamard matrices with maximum excess and classes with small excess, J. Combin. Des., 12, 4, 233-255, (2004) · Zbl 1045.05023
[8] Farmakis, Nikos; Kounias, Stratis, The excess of Hadamard matrices and optimal designs, Discrete Math., 67, 165-176, (1987) · Zbl 0652.05006
[9] Hammer, Joseph; Levingston, Richard; Seberry, Jennifer, A remark on the excess of Hadamard matrices and orthogonal designs, Ars Comb., 5, 237-254, (1978) · Zbl 0427.05019
[10] Holzmann, Wolfgang H.; Kharaghani, Hadi, On the excess of Hadamard matrices, Congr. Numer., 92, 257-260, (1993)
[11] Holzmann, Wolfgang H.; Kharaghani, Hadi; Lavassani, M. T., The excess problem and some excess inequivalent matrices of order \(32\), J. Stat. Plann. Inference, 72, 1-2, 381-391, (1998) · Zbl 0941.05015
[12] Kharaghani, Hadi, An infinite class of Hadamard matrices of maximal excess, Discrete Math., 89, 307-312, (1991) · Zbl 0734.05028
[13] Koukouvinos, Christos; Kounias, Stratis, Construction of some Hadamard matrices with maximum excess, Discrete Math., 85, 3, 295-300, (1990) · Zbl 0732.05016
[14] Koukouvinos, Christos; Kounias, Stratis; Seberry, Jennifer, Supplementary difference sets and optimal designs, Discrete Math., 88, 1, 49-58, (1991) · Zbl 0756.05032
[15] Koukouvinos, Christos; Seberry, Jennifer, Hadamard matrices of order \(≡ 8 \pmod {16}\) with maximal excess, Discrete Math., 92, 1-3, 173-176, (1991) · Zbl 0762.05024
[16] Kounias, Stratis; Farmakis, Nikos, On the excess of Hadamard matrices, Discrete Math., 68, 1, 59-69, (1988) · Zbl 0667.05013
[17] Lidl, Rudolf; Niederreiter, Harald, Finite Fields, 20, xiv+755 p. pp., (1997), Cambridge University Press · Zbl 0866.11069
[18] Meijer, Paul; van der Vlugt, Marcel, The evaluation of Gauss sums for characters of \(2\)-power order, J. Number Theory, 100, 2, 381-395, (2003) · Zbl 1030.11067
[19] Momihara, Koji; Suda, Sho, Conference matrices with maximum excess and two-intersection sets, Integers, 17, 15 p. pp., (2017)
[20] Muzychuk, Mikhail E., V-rings of permutation groups with invariant metric, (1987)
[21] Seberry, Jennifer, Existence of SBIBD\((4k^2,2k^2± k,k^2± k)\) and Hadamard matrices with maximal excess, Australas. J. Comb., 4, 87-91, (1991) · Zbl 0763.05016
[22] Storer, Thomas, Cyclotomy and Difference Sets, vii+134 p. pp., (1967), Markham Publishing Company · Zbl 0157.03301
[23] Xia, Tianbing; Xia, Mingyuan; Seberry, Jennifer, Regular Hadamard matrix, maximum excess and SBIBD, Australas. J. Comb., 27, 263-275, (2003) · Zbl 1027.05009
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