Set systems with kk-wise LL-intersections and codes with restricted Hamming distances

In this paper, we first give a corollary to Snevily’s Theorem on LL-intersecting families, which implies a result that cuts by almost half the bound given by Grolmusz and Sudakov (2002), and provide a kk-wise extension to the theorem by Babai et al. (2001) on set systems with LL-intersections modulo prime powers which implies polynomial bounds for such families. We then extend Alon–Babai–Suzuki type inequalities on set systems to kk-wise LL-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 ptpt, in contrast with Grolmusz’s result from Grolmusz (2006) that for non-prime power composite moduli, no polynomial bound exists for such codes.