Blocking sets and complete<i>k</i>-arcs

Abstract

Let be a finite projective plane of order n. Recall that a blocking set S in is a set of points which does not contain any line but which does intersect every line of . The first objective is to elaborate on the connection, pointed out by the writers, between blocking sets and complete A>arcs of . For example, the set of secants of a complete -arc with k < n + 2 dualizes to a blocking set. Using some simple observations, it is shown that a blocking set in a projective plane of order ten, if exists, contains at least 16 points. The proof uses a computer result on the nonexistence of complete 6-ares of due to R. H. F. Dennis ton. Using the result, a recent theorem concerning certain codes related to due to MacWilliams, Sloane, and Thompson is easily established. The result also shows that, in effect, a set of four mutually orthogonal latin squares of order ten is embeddable in a complete set in at most one way. This improves slightly on the bound of R. H. Bruck.