The excess of Hadamard matrices and optimal designs. (English) Zbl 0652.05006

Farmakis, Nikos; Kounias, Stratis

Discrete Math. 67, 165-176 (1987).

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

MSC:

05B20	Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B05	Combinatorial aspects of block designs

Keywords:

Hadamard matrix; maximal excess

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