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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 $$s^ 2t^ 2$$.

##### MSC:
 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.)
##### Keywords:
regular Hadamard matrix; excess; symmetric 2-design; 4-set