A 1-rotational (G,N,k,1) difference family is a set of k-subsets (base blocks) of an additive group G whose list of differences covers exactly 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-rotational over a group G if it admits G as an automorphism group fixing one point and acting regularly on the remainder. We prove that such a Steiner 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 k, an additive group G of order k − 1, and a prime power q ≡ 1 (mod k(k + 1)), a construction for 1-rotational (possibly resolvable) (G ⊕ Fq , G ⊕ {0}, k, 1) difference families is presented. This construction method always succeeds (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 [12] that has apparently been overlooked by several authors. Using our terminology, that construction appears to be a recursive construction for resolvable 1-rotational difference families over cyclic groups. Applying it in a particular case, we get a better result than previously known on cyclically resolvable 1-rotational (v, 4, 1)-BIBD’s.

G-invariantly resolvable Steiner 2-designs which are 1-rotational over G

BURATTI, Marco;
1998

Abstract

A 1-rotational (G,N,k,1) difference family is a set of k-subsets (base blocks) of an additive group G whose list of differences covers exactly 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-rotational over a group G if it admits G as an automorphism group fixing one point and acting regularly on the remainder. We prove that such a Steiner 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 k, an additive group G of order k − 1, and a prime power q ≡ 1 (mod k(k + 1)), a construction for 1-rotational (possibly resolvable) (G ⊕ Fq , G ⊕ {0}, k, 1) difference families is presented. This construction method always succeeds (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 [12] that has apparently been overlooked by several authors. Using our terminology, that construction appears to be a recursive construction for resolvable 1-rotational difference families over cyclic groups. Applying it in a particular case, we get a better result than previously known on cyclically resolvable 1-rotational (v, 4, 1)-BIBD’s.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/110243
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