The -ranks of residual and derived skew Hadamard designs

Let H be a Hadamard ( 4 n − 1 , 2 n − 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 ≠ 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 .