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SQUARE ROOTS OF DOUBLY REGULAR TOURNAMENT MATRICES
(Int Linear Algebra Soc, 2015) Hacioglu, Ilhan; Michael, T. S.; Ozdemir, Serhat
Fletcher asked whether there is a (0, 1)-matrix of order greater than 3 whose square is a regular tournament matrix. We give a negative answer for a special class of regular tournament matrices: T (0, 1)-matrix of order greater than 3 whose square is a doubly regular tournament matrix.
The p-ranks of residual and derived skew Hadamard designs
(Elsevier, 2011) Hacioglu, Ilhan; Michael, T. S.
Let H be a Hadamard (4n - 1, 2n - 1, n - 1)-design. Suppose that the prime p divides n, but that p(2) does not divide n. A result of Klemm implies that every residual design of H has p-rank at least n. Also, every derived design of H has p-rank at least n if p not equal 2. We show that when H is a skew Hadamard design, the p-ranks of the residual and derived designs are at least n even if p(2) divides n or p = 2. We construct infinitely many examples where the p-rank is exactly n. Published by Elsevier B.V.