Partitioning sets of triples into small planes

Mathon, Rudolf and Street, Anne Penfold (2002) Partitioning sets of triples into small planes. Designs Codes And Cryptography, 27 1-2: 119-130. doi:10.1023/A:1016506704066

Author Mathon, Rudolf
Street, Anne Penfold
Title Partitioning sets of triples into small planes
Journal name Designs Codes And Cryptography   Check publisher's open access policy
ISSN 0925-1022
Publication date 2002-10
Sub-type Article (original research)
DOI 10.1023/A:1016506704066
Volume 27
Issue 1-2
Start page 119
End page 130
Total pages 12
Editor D. Jungnickel
J.D. Key
S.A. Vanstone
Place of publication Boston, Mass., U.S.A.
Publisher Kluwer Academic Press
Collection year 2002
Language eng
Subject C1
230101 Mathematical Logic, Set Theory, Lattices And Combinatorics
780101 Mathematical sciences
Abstract We study partitions of the set of all ((v)(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 = 7(n) + 1, 13(n) + 1, 27(n) + 1, and affine partitions for v = 8(n) + 1, 9(n) + 1, 17(n) + 1. In particular, both Fano and affine partitions exist for v = 3(6n) + 1. Using properties of 3-wise balanced designs, we extend these results to show that affine partitions also exist for v = 3(2n). Similarly, mixed partitions are shown to exist for v = 8(n), 9(n), 11(n) + 1.
Keyword Computer Science, Theory & Methods
Mathematics, Applied
Triple Systems
Fano Partitions
Affine Partitions
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collections: Excellence in Research Australia (ERA) - Collection
School of Information Technology and Electrical Engineering Publications
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Created: Tue, 14 Aug 2007, 18:06:48 EST