Towards the determination of the regular $n$-covers of $PG(3,q)$

Oxenham, Martin ; Casse, Rey

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 57-87

Résumé

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 $q\geq n+2$.