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Hadamard matrices of order \(\equiv{} 8\)(mod 16) with maximal excess. (English) Zbl 0762.05024
The excess of a Hadamard matrix \(H\), denoted \(\sigma(H)\), is the sum of all its elements. \(\sigma(n)\) is used to denote the maximal excess of all Hadamard matrices of order \(n\). In this paper a new family of Hadamard matrices whose excess is maximal is given. It is shown that there is a Hadamard matrix of order \(4m(m-1)\) \((\equiv 0\pmod{16}\) and \(\equiv 8\pmod{16}\), respectively) whose excess meets the Kounias-Farmakis bound, i.e. \(\sigma(4m(m-1))=4(m-1)^ 2(2m+1)\).

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Full Text: DOI
[1] Best, M.R., The excess of a Hadamard matrix, Indag. math., 39, 357-361, (1977) · Zbl 0366.05016
[2] Enomoto, H.; Miyamoto, M., On maximal weights of Hadamard matrices, J. combin. theory ser. A, 29, 94-100, (1980) · Zbl 0445.05031
[3] Farmakis, N.; Kounias, S., The excess of Hadamard matrices and optimal designs, Discrete math., 67, 165-176, (1987) · Zbl 0652.05006
[4] Hammer, J.; Levingston, R.; Seberry, J., A remark on the excess of Hadamard matrices and orthogonal designs, Ars combin., 5, 237-254, (1978) · Zbl 0427.05019
[5] Jenkins, B.A.; Koukouvinos, C.; Kounias, S.; Seberry, J.; Seberry, R., Some results on the excesses of Hadamard matrices, Jcmcc, 4, 155-185, (1989) · Zbl 0713.05016
[6] H. Kharaghani, An infinite class of Hadamard matrices of maximal excess, Discrete Math., to appear. · Zbl 0734.05028
[7] C. Koukouvinos and S. Kounias, Construction of some Hadamard matrices with maximum excess, to appear. · Zbl 0732.05016
[8] Kounias, S.; Farmakis, N., On the excess of Hadamard matrices, Discrete math., 68, 59-69, (1988) · Zbl 0667.05013
[9] Mathon, R., Symmetric conference matrices of order pq2 + 1, Canad. J. math., 30, 321-331, (1978) · Zbl 0385.05018
[10] Sathe, Y.S.; Shenoy, R.G., Construction of maximal weight Hadamard matrices of order 48 and 80, Ars combin, 19, 25-35, (1985) · Zbl 0573.05013
[11] Schmidt, K.W.; Wang, E.T.H., The weights of Hadamard matrices, J. combin. theory ser. A, 23, 257-263, (1977) · Zbl 0428.05013
[12] Seberry, J., On skew Hadamard matices, Ars combin., 6, 255-275, (1978) · Zbl 0414.05008
[13] Seberry, J., SBIBD (4k2, 2k2 + k, k2 + k) and Hadamard matrices of order 4k2 with maximal excess are equivalent, Graphs combin., 5, 373-383, (1989)
[14] Seberry, J.; Whiteman, A.L., New Hadamard matrices and conference matrices obtained via Mathon’s construction, Graphs combin., 4, 355-377, (1988) · Zbl 0673.05016
[15] Wallis, W.D., On the weights of Hadamard matrices, Ars combin., 3, 287-292, (1977) · Zbl 0394.05010
[16] Yamada, M., On a series of Hadamard matrices of order 2^2t and the maximal excess of Hadamard matrices of order 22t, Graphs combin., 4, 297-301, (1988) · Zbl 0668.05013
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