AlonBabaiSuzuki's inequalities, FranklWilson type theorem and multilinear polynomials

Let K={k₁,k₂,⋯,k_r} and L={l ₁,l₂,⋯,l_s} be subsets of {0,1,⋯,p-1} such that K∩L=ø, where p is a prime. Let F={F ₁,F₂,⋯,F_m} be a family of subsets of [n]={1,2,⋯,n} with |F_i| (modp) ∈K for all F _i∈F and |F_i∩F_j| (modp) ∈L for any i≈j. Every subset F_i of [n] can be represented by a binary code a=(a₁,a₂,⋯,a_n) such that a_j=1 if j∈F_i and a_j=0 if j∉F_i. AlonBabaiSuzuki proved in non-modular version that if k_i≥s-r+1 for all i, then |F|≤∑_i=s-r+1^s(ni). We generalize it in modular version. AlonBabaiSuzuki also proved that the above bound still holds under r(s-r+1)≤p-1 and n<s+max_ik_i in modular version. AlonBabaiSuzuki made a conjecture that if they drop one condition r(s-r+1)≤p-1 among r(s-r+1)≤p-1 and n<s+max_ik_i, then the above bound holds. But we prove the same bound under dropping the opposite condition n<s+max_ik_i. So we prove the same bound under only condition r(s-r+1)≤p-1. This is a generalization of FranklWilson theorem (Frankl and Wilson, 1981 [2]).