Set systems with k-wise L-intersections and codes with restricted Hamming distances

In this paper, we first give a corollary to Snevily's Theorem on L-intersecting families, which implies a result that cuts by almost half the bound given by Grolmusz and Sudakov (2002), and provide a k-wise extension to the theorem by Babai et al. (2001) on set systems with L-intersections modulo prime powers which implies polynomial bounds for such families. We then extend Alon–Babai–Suzuki type inequalities on set systems to k-wise L-intersecting families and derive a result which improves the existing bound substantially for the non-modular case. We also provide the first known polynomial bounds for codes with restricted Hamming distances for all prime powers moduli p^t, in contrast with Grolmusz's result from Grolmusz (2006) that for non-prime power composite moduli, no polynomial bound exists for such codes.