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Resolutions of PG(5, 2) with point-cyclic automorphism group
Journal of Combinatorial Designs (1999)
  • Jumela F Sarmiento, Ateneo de Manila University
A t-(υ, k, λ) design is a set of υ points together with a collection of its k-subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d-dimensional projective geometry over GF(q), PG(d, q), is a 2-(qd + qd−1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2-(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = {R1, R2, …, Rs}, where s = (υ − 1)/(k−1) and each Ri consists of υ/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛσ = ℛ, then the design is said to be point-cyclically resolvable. The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point-transitive automorphism group. Furthermore, some necessary conditions for the point-cyclic resolvability of 2-(υ, k, 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000
Publication Date
December 14, 1999
Citation Information
Sarmiento, J. (2000). Resolutions of PG(5, 2) with point-cyclic automorphism group. Journal of Combinatorial Designs, 8(1), 2–14.<2::AID-JCD2>3.0.CO;2-H