×

zbMATH — the first resource for mathematics

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. \(\sigma(4m(m-1))=4(m-1)^ 2(2m+1)\).

MSC:
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.