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. Developing 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 $C_{kp}$-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 $C_{5p}$-invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1000.
On resolvable difference families
BURATTI, Marco
1997
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. Developing 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 $C_{kp}$-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 $C_{5p}$-invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1000.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
