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On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)

Abstract

We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound.

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Correspondence to Aaron C. S. Chan.

Additional information

A.C.S. Chan was supported during Summer 2008 at Simon Fraser University by NSERC of Canada via an Undergraduate Student Research Award.

J.A. Davis is supported by NSA Grant # MDA904-03-1-0032.

J. Jedwab is supported by NSERC of Canada.

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Chan, A.C.S., Davis, J.A. & Jedwab, J. On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7). J. Geom. 97, 29–44 (2010). https://doi.org/10.1007/s00022-010-0041-3

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  • DOI: https://doi.org/10.1007/s00022-010-0041-3

Mathematics Subject Classification (2010)

  • 05E20
  • 05B25
  • 94B05

Keywords

  • Centraliser
  • conjugacy class
  • Griesmer bound
  • integer linear program
  • linear code
  • prescribed automorphism group
  • projective code
  • projective set
  • rational canonical form
  • two-weightcode