On set systems with restricted intersections modulo and -ary -designs

We consider bounds on the size of families F of subsets of a v -set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections. We improve the known bound when F is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families F holds with equality, then F is the set of blocks of what we call a p -ary t -design for certain values of t . This motivates us to make a few observations about p -ary t -designs for their own sake.