The completion problem for partial packings
type: Item
instance of: scholarly article
title: The completion problem for partial packings
zbMATH Open document ID: 0566.51013
DOI: 10.1007/BF00181225
author: Gary L. Ebert
published in: Geometriae Dedicata
publication date: 1985-01-01T00:00:00Z
review text: Denote the three-dimensional projective space over the finite Galois field GF(q) by \(\Sigma =PG(3,q)\). A spread of \(\Sigma\) is a collection of \(q^ 2+1\) pairwise skew lines of \(\Sigma\). A packing of \(\Sigma\) is a collection of \(q^ 2+q+1\) spreads, no two of which have a line in common. A partial packing is any collection of pairwise disjoint spreads of \(\Sigma\). Line packings of \(\Sigma\) were the first packings studied but it is an open question as to when a partial packing can be completed to a packing in this case. This paper shows that there is no certainty, combinatorially, of completing such a packing even when the deficiency is 2. Using primitive elements of finite fields, the author constructs, for every odd prime power of q, a set of \(2(q^ 2+1)\) lines which doubly cover the points of \(\Sigma\) but which cannot be partitioned into two spreads.
reviewed by: A. F. Horadam
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 3903436
zbMATH Keywords: completion problem for partial packings
zbMATH Keywords: finite projective space
zbMATH Keywords: proper 2-cover
zbMATH Keywords: ovoid
zbMATH Keywords: fibration
zbMATH Keywords: spread
zbMATH Keywords: deficiency
zbMATH Keywords: primitive elements
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3903436