### 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 q^{2} + 1 + (q - 1)/3 lines. We also show that Q(4, q), q even, does not contain minimal covers of sizes q^{2} + 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, q^{2}) arising from the Hermitian variety in PG(4, q^{2}). It is known that U(4, q^{2}) does not have an ovoid. Here, we show that it also does not contain minimal blocking sets of sizes q^{5} + 2, q^{5} + 3 and q^{5} + 4 except maybe for small values of q.

Original language | English |
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Pages (from-to) | 35-51 |

Number of pages | 17 |

Journal | Discrete Mathematics |

Volume | 238 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jul 28 2001 |

Event | The Third Shanghai Conference - Shanghai, China Duration: May 15 1999 → May 19 1999 |

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### Keywords

- Blocking sets
- Covers
- Generalized quadrangles

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*238*(1-3), 35-51. https://doi.org/10.1016/S0012-365X(00)00418-0