G-INVARIANTLY RESOLVABLE STEINER 2-DESIGNS WHICH ARE 1-ROTATIONAL OVER-G

A 1-rotational (G, N, k, 1) difference family is a set of k-subsets (b ase blocks) of an additive group G whose list of differences covers ex actly once G - N and zero times N, N being a subgroup of G of order k - 1. We say that such a difference family is resolvable when the base blocks union is a system of representatives for the nontrivial right ( or left) cosets of N in G. A Steiner 2-design is said to be 1-rotation al over a group G if it admits G as an automorphism group fixing one p oint and acting regularly on the remainder. We prove that such a Stein er 2-design is G-invariantly resolvable (i.e. it admits a G-invariant resolution) if and only if it is generated by a suitable 1-rotational resolvable difference family over G. Given an odd integer Ic, an addit ive group G of order k - 1, and a prime power q = 1 (mod k(k + 1)), a construction for 1-rotational (possibly resolvable) (G+IFq, G+{0}, k, 1) difference families is presented. This construction method always s ucceeds (resolvability included) for k = 3. For small values of k > 3, the help of a computer allows to find some new 1-rotational (in many cases resolvable) ((k - 1)q + 1, k, 1)-BIBD's. In particular, we find (1449, 9, 1) and (4329, 9, 1)-BIBD's the existence of which was still undecided. Finally, we revisit a construction by Jimbo and Vanstone [1 2] that has apparently been overlooked by several authors. Using our t erminology, that construction appears to be a recursive construction f or resolvable 1-rotational difference families over cyclic groups. App lying it in a particular case, we get a better result than previously known on cyclically resolvable 1-rotational (v, 4, 1)-BIBD's.