Covers and blocking sets of classical generalized quadrangles

J. Eisfeld, L. Storme, T. Szonyi, P. Sziklai

Research output: Contribution to journalConference article

7 Citations (Scopus)

Abstract

This article discusses two problems on classical generalized quadrangles. It is known that the generalized quadrangle Q(4, q) arising from the parabolic quadric in PG(4, q) has a spread if and only if q is even. Hence, for q odd, the problem arises of the cardinality of the smallest set of lines of Q(4, q) covering all points of Q(4, q). We show in this paper that this set of lines must contain more than q2 + 1 + (q - 1)/3 lines. We also show that Q(4, q), q even, does not contain minimal covers of sizes q2 + 1 + r when q ≥ 32 and 0 < r ≤ √q. To obtain this latter result, we generalize a result on minimal covers of lines in PG(3, q) to minimal covers of lines of the classical generalized quadrangles. This result is then also used to study minimal blocking sets of the non-singular generalized quadrangle U(4, q2) arising from the Hermitian variety in PG(4, q2). It is known that U(4, q2) does not have an ovoid. Here, we show that it also does not contain minimal blocking sets of sizes q5 + 2, q5 + 3 and q5 + 4 except maybe for small values of q.

Original languageEnglish
Pages (from-to)35-51
Number of pages17
JournalDiscrete Mathematics
Volume238
Issue number1-3
DOIs
Publication statusPublished - Jul 28 2001
EventThe Third Shanghai Conference - Shanghai, China
Duration: May 15 1999May 19 1999

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Keywords

  • Blocking sets
  • Covers
  • Generalized quadrangles

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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