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Existence of SBIBD (\(4k^ 2,2k^ 2\pm k,k^ 2\pm k\)) and Hadamard matrices with maximal excess. (English) Zbl 0763.05016
The authoress continues her investigation on the existence of symmetric \(2-(4k^ 2,2k^ 2-k,\;k^ 2-k)\) designs and regular Hadamard matrices of order \(4k^ 2\) with maximal excess \(8k^ 2\). For her earlier results see, for instance, J. Seberry and A. L. Whiteman [Graphs Comb. 4, No. 4, 355-377 (1988; Zbl 0673.05016)] and J. Seberry [Graphs Comb. 5, No. 4, 373-383 (1989; Zbl 0713.05017)]. The list of \(k\)’s for which she proved the existence is a little too complicated to be presented here. Instead, an auxiliary concept of a regular 4-set of regular matrices of order \(k^ 2\) and a theorem concerning it are given here. (If there exists such a 4-set, then the existence of a symmetric 2- design and a regular Hadamard matrix given above follow.) A complete regular 4-set of regular matrices of order \(k^ 2\) is a set of four symmetric \((-1,1)\)-matrices \(A_ 1,A_ 2,A_ 3\) and \(A_ 4\) of order \(k^ 2\) such that \(\sum^ 4_{i=1}A_ i^ 2=4k^ 2I\), \(A_ iJ=kJ\) and \(A_ iA_ j=aJ\), where \(i,j=1,2,3,4\), \(i\neq j\), \(a\) is a constant, and \(I\) and \(J\) denote the identity and all one matrices respectively. Theorem 5. If there exist such 4-sets of orders \(s^ 2\) and \(t^ 2\), then there exists such a 4-set of order \(s^ 2t^ 2\).

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B05 Combinatorial aspects of block designs
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)