Linear Systems over Finite Abelian Groups

We consider a system of linear constraints over any finite Abelian group G of the following form: ℓ_i(x₁, ..., x_n) ≡ ℓ_i,1x₁ + ⋯ + ℓ_i,nx_n ∈ A_i for i=1, ..., N and each A_i ⊂ G, ℓ_i,j is an element of G and x_i´s are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MOD_q boolean function, when the order of G and q are co-prime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS´09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential bounds on the size of boolean depth-four circuits of the form MAJ ο AND ο ANY ο₍₁₎ ο MOD_m for computing the MOD_q function, when m, q are co-prime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity´97).