Small proper double blocking sets in Galois planes of prime order

Abstract

A proper double blocking set in PG ( 2 , p ) is a set B of points such that 2 ⩽ | B ∩ l | ⩽ ( p + 1 ) - 2 for each line l. The smallest known example of a proper double blocking set in PG ( 2 , p ) for large primes p is the disjoint union of two projective triangles of side ( p + 3 ) / 2 ; the size of this set is 3 p + 3 . For each prime p ⩾ 11 such that p ≡ 3 ( mod 4 ) we construct a proper double blocking set with 3 p + 1 points, and for each prime p ⩾ 7 we construct a proper double blocking set with 3 p + 2 points.