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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

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