Difference sets in abelian groups of \(p\)-rank two
A \((v, k, \lambda)\)-difference set \(D\) is a \(k\)-subset of a group \(G\) such that the list of differences covers every non-identity group element exactly \(\lambda\) times. One of the open problems in finite geometry is to decide for which parameter triples difference sets can exist. In the case of abelian groups, a powerful approach to prove non-existence results is the use of algebraic number theory. Using this method, one can develop rather strong results if a Sylow \(p\)-subgroup of \(G\) is cyclic. In this paper, the authors study the case of Sylow \(p\)-subgroups of rank 2. In particular, they prove the non-existence of (96, 20, 4)-difference sets in \(\mathbb{Z}_ 4\times \mathbb{Z}_ 8\times \mathbb{Z}_ 3\), (640, 72, 8)- difference sets in \(\mathbb{Z}_ 8\times \mathbb{Z}_{16}\times \mathbb{Z}_ 5\), and (320, 88, 24)-difference sets in \(\mathbb{Z}_ 8\times \mathbb{Z}_ 8\times \mathbb{Z}_ 5\). Recently, a new difference set with parameters (96, 20, 4) in \(\mathbb{Z}_ 4\times \mathbb{Z}_ 4\times \mathbb{Z}_ 2\times \mathbb{Z}_ 3\) has been constructed by the same authors [J. Comb. Theory, Ser. 69, No. 1, 170-172 (1995; see the immediately preceding item Zbl 0822.05004)].
scientific article; zbMATH DE number 722114
scientific article