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

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