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Partitioning Sets of Triples into Small Planes

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Abstract

We study partitions of the set of all v3 triples chosen from a v-set intopairwise 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 severalcases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We constructsuch partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, andan affine partition for v = 18. Using these as starter partitions, we prove that Fano partitionsexist 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 partitionsexist for v = 36n + 1. Using properties of 3-wise balanced designs, weextend 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.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, M5S 1A4

    Rudolf Mathon

  2. Centre for Discrete Mathematics and Computing, The University of Queensland, Brisbane, Australia, 4072

    Anne Penfold Street

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  1. Rudolf Mathon
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  2. Anne Penfold Street
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Mathon, R., Street, A.P. Partitioning Sets of Triples into Small Planes. Designs, Codes and Cryptography 27, 119–130 (2002). https://doi.org/10.1023/A:1016506704066

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