Sets of disjoint lines in ${ m PG}(3,,q)$

Dale M. Mesner
1967 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
1. Spreads and partial spreads. Let S be a projective space PG(3, q) of dimension 3 and finite order q. Then 2 contains (q + l)(q 2 + 1) points and an equal number of planes, and (q 2 + 1) (q 2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q 2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is
more » ... Each point is contained in q 2 + q + 1 planes and an equal number of lines. A spread of lines of 2 is a set of q 2 + 1 lines of 2 which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines. A packing of lines in 2 is a set of q 2 + q + 1 spreads such that every line is in exactly one spread of the set. Spreads of lines exist in every PG(3, q) and packings are known to exist in some cases. A more general concept is that of a spread or packing of disjoint PG(m -1, g)'s, subspaces of dimension m -1, in PG (Vz -1, q) , where m is a divisor of n. Generalizations to infinite geometries can also be formulated. All spreads and packings mentioned in this paper are to be taken as spreads and packings of lines in a finite 3-space. The terminology and the recent theory of spreads are due to R. H. Bruck (1;2), although related ideas have been considered earlier. A linear congruence (3) in 2 is a special case of a spread. A finite geometry which admits a packing is an instance of a balanced incomplete block design which is resolvable. Example. The 15 points of PG(3, 2) may be represented as follows by coordinate vectors over GF(2): Each of the 35 lines of this geometry contains 3 points as displayed in one row of Table I . Each section of the table contains a spread, a set of 5 disjoint lines whose union is the set of all 15 points. The seven spreads listed contain each line exactly once and therefore comprise a packing. The points of PG(3, 4) given in the table are for later use.
doi:10.4153/cjm-1967-019-5 fatcat:2t3fqv3lljhlbb6zg5cfxhwsgy