zbMATH — the first resource for mathematics

The excess of Hadamard matrices and optimal designs. (English) Zbl 0652.05006
The authors are interested in maximizing the determinant of a matrix R of the following form: \(R=\left( \begin{matrix} 1\\ e\end{matrix} \begin{matrix} -e\quad T\\ H\end{matrix} \right)\), where H is an Hadamard matrix, e is the all one vector of size (n,1) and T denotes the transposition. It turns out that they should construct an Hadamard matrix of maximal excess. Let s(H) be the excess of G and \(s(n)=\max s(H)\), where H runs over all Hadamard matrices of order n. In this paper they obtain that \(s(40)=244\), \(s(44)=280\), \(s(48)=324\), \(s(52)=364\), \(s(80)=704\), and \(s(84)=756\). Moreover for \(n=40\) and 44 they obtain patterns for row and column-sum vectors of H such that \(s(H)=s(n)\).
Reviewer: N.Ito

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B05 Combinatorial aspects of block designs
Full Text: DOI
[1] Best, M.R, The excess of a Hadamard matrix, Indag. math., 39, 357-361, (1977) · Zbl 0366.05016
[2] Chadjipantelis, T; Kounias, S; Moyssiadis, C, The maximum determinant of 21×21 (+1, −1)-matrices and D-optimal designs, J. statist. plann. inference, (1987), to appear · Zbl 0625.62062
[3] Enlich, H, Determinantenabschätzungen für binäre matrizen, Math. zeitsch., 83, 123-132, (1964) · Zbl 0115.24704
[4] Enomoto, H; Miyamoto, M, On maximal weights of Hadamard matrices, J. combin. theory ser. A, 29, 94-100, (1980) · Zbl 0445.05031
[5] Moyssiadis, C; Kounias, S, The exact D-optimal first order saturated design with 17 observations, J. statist. plann. inference, 7, 13-27, (1982) · Zbl 0515.62072
[6] Raghavarao, D, Some optimum weighing designs, Ann. math. statist., 30, 295-303, (1959) · Zbl 0097.13504
[7] 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
[8] Schmidt, K.W; Wang, E.T.H, The weights of Hadamard matrices, J. combin. theory ser. A, 23, 257-263, (1977) · Zbl 0428.05013
[9] T.V. Trung, The existence of symmetric block designs with parameters (41, 16, 6) and (66, 26, 10), J. Combin. Theory Ser. A 33, 201-204. · Zbl 0519.05008
[10] Wallis, W.D, On the weights of Hadamard matrices, Ars combin., 3, 287-292, (1977) · Zbl 0394.05010
[11] Wallis, W.D; Street, A.P; Seberry Wallis, J, Combinatorics: room squares, sum-free sets, Hadamard matrices, () · Zbl 1317.05003
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.