Largest minimal blocking sets in PG(2,8)

Abstract

Bruen and Thas proved that the size of a large minimal blocking set is bounded by q Á ffiffi ffi q p þ 1. Hence, if q ¼ 8, then the maximal possible size is 23. Since 8 is not a square, it was conjectured that a minimal blocking 23-set does not exist in PGð2; 8Þ. We show that this is not the case, and construct such a set. We prove that this is combinatorially unique. We also complete the spectrum problem of minimal blocking sets for PGð2; 8Þ by showing a minimal blocking 22-set. #

Classification of minimal blocking sets in small Desarguesian projective planes

“…In [1] Barát and Innamorati construct the unique (up to equivalence) minimal blocking set of size 23 and prove its uniqueness. They also construct the minimal blocking set of size 22 but did not establish its uniqueness.…”

On the spectrum of minimal blocking sets in PG$(2,q)$

“…There is no minimal blocking set of size 14 (see Barát, Del Fra, Innamorati, Storme [5]), the IMI construction gives a minimal blocking set for each size between 15 and 21. Recently, Barát and Innamorati [4] constructed minimal blocking sets of size 22, 23, and 23 is the Bruen-Thas upper bound. The construction is very nice for 23, the blocking set consists of three disjoint Fano subplanes from a subplane partition of the plane minus a simplex, plus two vertices of the simplex.…”

On large minimal blocking sets in PG(2,q)

“…The case q ¼ 8 was recently studied in detail by Barát and Innamorati [2], who constructed a minimal blocking set of size 23 in PGð2; 8Þ. Note that 23 is just the integer part of 8 ffiffi ffi 8 p þ 1, that is, the Bruen-Thas upper bound is sharp for q ¼ 8.…”