Nested Hadamard difference sets
An Hadamard difference set (HDS) is a \((4n^2, 2n^2- n, n^2- n)\)-difference set. Hadamard difference sets on direct products of cyclic groups of prime-power orders are of particular interest due to their applications in signal processing, and have recently been studied by several authors. The main focus of the article is on Hadamard difference sets on abelian groups of the form \(H\times{\mathcal Z}^2_{p^\alpha}\), \(H\) a direct product of cyclic 2-groups, \(p\) a prime, and, in particular, the question of nonexistence of Hadamard difference sets for some of these groups. Using the correspondence between HDSs and perfect binary arrays together with the idea of a contraction by a subgroup, the authors were able to prove that an HDS on \(H\times{\mathcal Z}^2_{p^\alpha}\) implies the existence of an HDS for all \(H\times{\mathcal Z}^2_{p^k}\), \(1\leq k\leq\alpha\), provided the prime \(p\) is semiprimitive mod the exponent of \(H\). It follows that in such a case the nonexistence of an HDS on \(H\times{\mathcal Z}^2_p\) implies the nonexistence of HDSs for all \(H\times{\mathcal Z}^2_{p^k}\), \(k\geq 1\). This allowed them to show the nonexistence of HDSs for the groups \(H\times{\mathcal Z}^2_{7^k}\) and \(H\times{\mathcal Z}^2_{11^k}\), where \(H\) is any group of order 64 and exponent \(\leq 8\) and \(k\geq 1\).
scientific article; zbMATH DE number 1057361
scientific article