# 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, RudolfStreet, Anne Penfold Partitioning sets of triples into small planes Designs Codes And Cryptography   Check publisher's open access policy 0925-1022 2002-10 Article (original research) 10.1023/A:1016506704066 27 1-2 119 130 12 D. JungnickelJ.D. KeyS.A. Vanstone Boston, Mass., U.S.A. Kluwer Academic Press 2002 eng C1230101 Mathematical Logic, Set Theory, Lattices And Combinatorics780101 Mathematical sciences 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. Computer Science, Theory & MethodsMathematics, AppliedPartitionsTriple SystemsFano PartitionsAffine PartitionsSystems C1

 Document type: Journal Article Article (original research) Excellence in Research Australia (ERA) - Collection School of Information Technology and Electrical Engineering Publications

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