On Resolvable Difference Families

Abstract

A Steiner 2-design is said to be G-invariantly resolvable if admits an automorphism group G and a resolution invariant under G. Introducing and studying resolvable difference families, we characterize the class of G-invariantly resolvable Steiner 2-designs arising from relative difference families over G. Such designs have been already studied by Genma, Jimbo, and Mishima [13] in the case in which G is cyclic. Developping their results, we prove that any (p, k, 1)-DF (p prime) whose base blocks exactly cover p−1/k(k−1) distinct cosets of the k-th roots of unity (mod p), leads to a Ckp-invariantly resolvable cyclic (kp,k,1)-BBD. This induced us to propose several constructions for DF‘s having this property. In such a way we prove, in particular, the existence of a C5p-invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1.000.