IngentaConnect Partitioning Sets of Triples into Small Planes

Authors: Mathon R.¹; Street A.P.²

Source: Designs, Codes and Cryptography, Volume 27, Numbers 1-2, October 2002 , pp. 119-130(12)

Publisher: Springer

Abstract:

We study partitions of the set of all {v}\choose{3} triples chosen from a v-set into pairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2, 2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions)or copies of some planes of each type (mixed partitions).

We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in several cases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We construct such partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, and an affine partition for v = 18. Using these as starter partitions, we prove that Fano partitions exist for v = 7ⁿ + 1, 13ⁿ + 1, 27ⁿ + 1, and affine partitions for v = 8ⁿ + 1, 9ⁿ + 1, 17ⁿ + 1. In particular, both Fano and affine partitions exist for v = 3⁶ⁿ + 1. Using properties of 3-wise balanced designs, we extend these results to show that affine partitions also exist for v = 3²ⁿ.

Similarly, mixed partitions are shown to exist for v = 8ⁿ, 9ⁿ, 11ⁿ + 1.

Keywords: partitions; triple systems; Fano partitions; affine partitions

Language: English

Document Type: Research article

Affiliations: 1: Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 1A4 2: Centre for Discrete Mathematics and Computing, The University of Queensland, Brisbane, Australia 4072

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