Blocking and double blocking sets in finite planes
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Jan De Beule, Tamás Héger, Tamás Sz˝, Geertrui Van De Voorde
2016 Volume 23
Abstract
In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q 2 of size q 2 + 2q + 2 admitting * This author has been supported as a postdoctoral fellow of the Research Foundation Flanders (Bel-gium) (FWO). 1 1-,2-,3-,4-, (q + 1)-and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q 2 of size at most 4q 2 /3 + 5q/3, which is considerably smaller than 2q 2 − 1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q 2. We also consider particular André planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.
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