The \(p\)-ranks of skew Hadamard designs (Q1906128)

The incidence matrix of a Hadamard \((4n- 1, 2n- 1, n- 1)\)-design \(D\) is a \((4n- 1)\times (4n- 1)\) \((0, 1)\)-matrix \(A\) that satisfies \(AA^T= nI+ (n- 1) J\). An Hadamard design is skew provided it is isomorphic to a design whose incidence matrix \(A\) satisfies \(A+ A^T+ I= J\) after row and column permutations. The \(p\)-rank of a design \(D\), denoted \(\text{rank}_p(D)\), is the rank of its incidence matrix over a field of characteristic \(p\). \textit{D. Jungnickel} [Difference sets, Contemporary design theory, Collect. Surv., 241-324 (1992; Zbl 0768.05013)] proved that \(\text{rank}_p(D)= 2n\) whenever \(D\) is a skew Hadamard design arising from a difference set in an abelian group of prime power order. This character-theoretic argument is borrowed from \textit{F. J. MacWilliams} and \textit{H. B. Mann} [Inf. Control 12, 474-488 (1968; Zbl 0169.32104)]. The author generalizes and simplifies this result and proves that if \(p\) divides \(n\), then the \(p\)-rank of a skew Hadamard \((4n- 1, 2n- 1, n- 1)\)-design is \(2n\).