Vol. 53, No. 1, 1974

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Blocking sets and complete k-arcs

Aiden A. Bruen and J. C. Fisher

Vol. 53 (1974), No. 1, 73–84

Let π be a finite projective plane of order n. Recall that a blooking set lS in π is a set of points which does not contain any line but which does intersect every line of π. The first objective is to elaborate on the connection, pointed out by the writers, between blocking sets and complete k-arcs of π. For example, the set of secants of a complete k-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 π of order ten, if π exists, contains at least 16 points. The proof uses a computer result on the nonexistence of complete 6-arcs of π due to R. H. F. Denniston. Using the result, a recent theorem concerning certain codes related to π 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.

Mathematical Subject Classification 2000
Primary: 05B25
Received: 3 July 1973
Published: 1 July 1974
Aiden A. Bruen
J. C. Fisher