Minimal blocks of points with weight divisible by p over GF(p)

We construct three families of minimal blocks over GF(p) where p is an odd prime. For example, we show that the points in rank-(2p−1) projective space PG(2p−2,p) with p coordinates equal to 1 and p−1 coordinates equal to 0 form a minimal 1-block over GF(p). The proofs use the Chevalley–Warning theorem about the number of zeros of polynomials over finite fields.