×

zbMATH — the first resource for mathematics

On the excess of Hadamard matrices. (English) Zbl 0667.05013
Bounds for the excess \(\sigma\) (H) of a Hadamard matrix H the sum of the elements of the matrix H, are established in the work. \(\sigma\) (n) denotes the maximum of \(\sigma\) (H) for all Hadamard matrices of order n. Upper bounds for \(\sigma\) (n), with n from various intervals, are obtained, improving the well-known bound \(n\sqrt{n}\). The established bound coincides with \(n\sqrt{n}\) when \(n=(4m)^ 2\) or \(n=4(2m+1)^ 2\) and the matrix H is regular, i.e. it has all row and column sums equal.
Regular matrices are constructed for \(n=4(2m+1)^ 2\) when \(m=1,2,3\) and their construction for \(m=4,9,12\) is indicated. A procedure is described to find all row-sum and column-sum vectors of a Hadamard matrix with given excess.
Reviewer: S.S.Agayan

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] Chadjipantelis, T.; Kounias, S., Supplementary difference sets and D-optimal designs for n≡2 (mod 4), Discrete math., 57, 211-216, (1985) · Zbl 0589.05020
[3] Enomoto, H.; Miyamoto, M., On maximal weights of Hadamard matrices, J. combin. theory ser. A, 29, 94-100, (1980) · Zbl 0445.05031
[4] Farmakis, N.; Kounias, S., The excess of Hadamard matrices and optimal designs, Discrete math., 67, 2, 165-176, (1985), (1987) · Zbl 0652.05006
[5] Goethals, J.M.; Seidel, J.J., A skew-Hadamard matrix of order 36, J. austral. math. soc., 11, 343-344, (1970) · Zbl 0226.05015
[6] 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
[7] Schmidt, K.W.; Wang, E.T.H., The weights of Hadamard matrices, J. combin. theory ser A, 23, 257-263, (1977) · Zbl 0428.05013
[8] Wallis, W.D.; Street, A.P.; Wallis, J.S., Combinatorics: room squares, sum-free sets, Hadamard matrices, Vol. 292, (1972), Springer Berlin, Lecture Notes in Mathematics · Zbl 1317.05003
[9] Wallis, W.D., On the weights of Hadamard matrices, Ars combin., 3, 287-292, (1977) · Zbl 0394.05010
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.