Small proper double blocking sets in Galois planes of prime order

A proper double blocking set in PG(2,p)PG(2,p) is a set B   of points such that 2⩽|B∩l|⩽(p+1)-22⩽|B∩l|⩽(p+1)-2 for each line l  . The smallest known example of a proper double blocking set in PG(2,p)PG(2,p) for large primes p   is the disjoint union of two projective triangles of side (p+3)/2(p+3)/2; the size of this set is 3p+33p+3. For each prime p⩾11p⩾11 such that p≡3(mod4) we construct a proper double blocking set with 3p+13p+1 points, and for each prime p⩾7p⩾7 we construct a proper double blocking set with 3p+23p+2 points.