The completion problem for partial packings
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.
scientific article; zbMATH DE number 3903436