Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Jan De Beule, Patrick Govaerts, Anja Hallez, L. Storme

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q−(5,3), to 56-caps contained in this elliptic quadric Q−(5,3).
Original languageEnglish
Pages (from-to)187-201
JournalDesigns, Codes and Cryptography
Volume50
Issue number2
DOIs
Publication statusPublished - Feb 2009

Keywords

  • Tight sets
  • m-Covers
  • m-Ovoids
  • Minihypers
  • Polar spaces
  • Generalized quadrangles

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