The p-ranks of residual and derived skew Hadamard designs

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.