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.

Abstract (VO)

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 $(n\ge 3)$ exist whenever $q\ge n+2$.

Riassunto

Si dice che un insieme $S$ di rette di $PG(3,q)$ copre $n$ volte un punto $P$ di $PG(3,q)$, se esistono esattamente $n$ rette di $S$ incidenti $P$. Un insieme di rette di $PG(3,q)$ che copre $n$ volte ogni punto di $PG(3,q)$ si dice $n$-cover. In questa nota, dopo una descrizione degli esempi noti di $n$-cover e delle rispettive proprietà, viene mostrato come gli $n$-cover di $PG(3,q)$ possono essere utilizzati per la costruzione di classi di disegni di Sperner quasi-$n$-multipli. Infine, allo scopo di ottenere nuovi esempi di tali disegni mediante la derivazione di quelli esistenti, si introduce la nozione di n-cover regolare. I risultati principali sono: la dimostrazione della non esistenza di un $2$-cover regolare di $PG(3,q)$ per $q>2$ e quella della non esistenza di un $n$-cover regolare $(n\ge 3)$ per $q\ge n+2$.

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