The spectrum of minimal blocking sets

Abstract

Let S( q) denote the spectrum of minimal blocking sets in a projective plane of order q. Innamorati and Maturo (Ratio Math. 2 (1991) 151–155) proved that if q⩾4 then [2 q−1,3 q−5]∪{3 q−3}⊆ S( q) and if the plane is Desarguesian then [2 q−1,3 q−3]⊆ S( q). The spectral problem remains to be solved, see Blokhuis (Bull. London Math. Soc. 18 (1986) 132–134); the object of this paper is to study the existence and the uniqueness of certain situations. Several constructions which permit to obtain minimal blocking sets modifying known examples are presented. Moreover, a combinatorial technique to prove the uniqueness of certain configurations realizing largest minimal blocking sets is introduced. The method is applied to the first open case: the uniqueness of a minimal blocking 19-set in PG(2,7).