We consider a system of linear constraints over any finite Abelian group G of the following form: \ell_i(x_1,\ldots,x_n) \equiv \ell_{i,1}x_1+\cdots+\ell_{i,n}x_n \in A_i for i=1,\ldots,t and each A_i \subset G, \ell_{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 \text{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 \text{MAJ} \circ \text{AND} \circ \text{ANY}_{O(1)} \circ \text{MOD}_m for computing the \text{MOD}_q function, when m,q are co-prime. No superpolynomial lower bounds were known for such circuits for computing any explicit function.%, when m had three or more distinct prime factors.

This completely solves an open problem posed by Beigel and Maciel (Complexity'97).