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 p2 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≠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 p2 divides n or p=2. We construct infinitely many examples where the p-rank is exactly n.