New upper bounds for the size of set systems with restricted intersections modulo prime powers

Let $q=p^{\alpha }$ be a fixed prime power, $k\ge 2$ be an integer. We give a new upper bound for the size of $k$-wise $q$-modular $L$-avoiding $L$-intersecting set systems, where $L$ is any proper subset of $\{0,\dots ,q-1\}$. Our proof is based on the linear algebra bound method and basic number theory.