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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. [ABSTRACT FROM AUTHOR]