Small 3-fold blocking sets in PG(2,p^n)

Csajbók, Bence and Kepes, Máté Róbert and Robin, Eszter Melinda and Sógor, Bence and Wang, Sherry and Williams, Elias (2026) Small 3-fold blocking sets in PG(2,p^n). EUROPEAN JOURNAL OF COMBINATORICS, 137. No. 104414. ISSN 0195-6698

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Abstract

A t-fold blocking set of the finite Desarguesian plane PG(2,p^n), p prime, is a set of points meeting each line of the plane in at least t points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. 1-fold blocking sets not containing a line) in PG(2,p^n) is an open question when n ≥ 5 is odd and 3 ∤ n. For n > 1 the conjectured lower bound for this size is (p^n + p^n(s−1)/s + 1), where p^(n/s) is the size of the largest proper subfield of F_p^n. Since the union of t pairwise disjoint nontrivial blocking sets is a t-fold blocking set, it is conjectured that when pn/s is large enough w.r.t. t, then the minimum size of a t-fold blocking set in PG(2,p^n) is t(p^n + p^n(s−1)/s + 1). If n is even, then the decomposition of the plane into disjoint Baer subplanes gives a t-fold blocking set of this size. However, for odd n, the existence of such sets is an unsolved problem in most cases.

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