Towards the determination of the regular n -covers of P ⁢ G ⁢ 3 , q

Martin Oxenham; Rey Casse

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 1, page 57-87
  • ISSN: 0392-4041

Abstract

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A set of lines S of P ⁢ G ⁢ 3 , q is said to cover a point P of P ⁢ G ⁢ 3 , q n times if there are exactly n lines of S incident with P . An n -cover of P ⁢ G ⁢ 3 , q is a set of lines of P ⁢ G ⁢ 3 , q which covers each point of P ⁢ G ⁢ 3 , q n times. In this paper, the properties and known examples of n -covers are reviewed and it is demonstrated how n -covers of P ⁢ G ⁢ 3 , q can be used to construct classes of quasi- n -multiple Sperner designs. Finally, motivated by the problem of deriving these designs to arrive at new examples, the notion of regular n -covers of P ⁢ G ⁢ 3 , q is introduced. The main results of the paper are that no regular 2 -covers of P ⁢ G ⁢ 3 , q exist for q > 2 and that no regular n -covers n ≥ 3 exist whenever q ≥ n + 2 .

How to cite

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Oxenham, Martin, and Casse, Rey. "Towards the determination of the regular $n$-covers of $PG(3,q)$." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 57-87. <http://eudml.org/doc/196151>.

@article{Oxenham2003,
abstract = {A set of lines $S$ of $PG(3, q)$ is said to cover a point $P$ of $PG(3, q)$$n$ times if there are exactly $n$ lines of $S$ incident with $P$. An $n$-cover of $PG(3, q)$ is a set of lines of $PG(3, q)$ which covers each point of $PG(3, q)$$n$ times. In this paper, the properties and known examples of $n$-covers are reviewed and it is demonstrated how $n$-covers of $PG(3, q)$ can be used to construct classes of quasi-$n$-multiple Sperner designs. Finally, motivated by the problem of deriving these designs to arrive at new examples, the notion of regular $n$-covers of $PG(3, q)$ is introduced. The main results of the paper are that no regular $2$-covers of $PG(3, q)$ exist for $q>2$ and that no regular $n$-covers $(n\geq 3)$ exist whenever $q\geq n+2$.},
author = {Oxenham, Martin, Casse, Rey},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {57-87},
publisher = {Unione Matematica Italiana},
title = {Towards the determination of the regular $n$-covers of $PG(3,q)$},
url = {http://eudml.org/doc/196151},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Oxenham, Martin
AU - Casse, Rey
TI - Towards the determination of the regular $n$-covers of $PG(3,q)$
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/2//
PB - Unione Matematica Italiana
VL - 6-B
IS - 1
SP - 57
EP - 87
AB - A set of lines $S$ of $PG(3, q)$ is said to cover a point $P$ of $PG(3, q)$$n$ times if there are exactly $n$ lines of $S$ incident with $P$. An $n$-cover of $PG(3, q)$ is a set of lines of $PG(3, q)$ which covers each point of $PG(3, q)$$n$ times. In this paper, the properties and known examples of $n$-covers are reviewed and it is demonstrated how $n$-covers of $PG(3, q)$ can be used to construct classes of quasi-$n$-multiple Sperner designs. Finally, motivated by the problem of deriving these designs to arrive at new examples, the notion of regular $n$-covers of $PG(3, q)$ is introduced. The main results of the paper are that no regular $2$-covers of $PG(3, q)$ exist for $q>2$ and that no regular $n$-covers $(n\geq 3)$ exist whenever $q\geq n+2$.
LA - eng
UR - http://eudml.org/doc/196151
ER -

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