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2011 IEEE 26th Annual Conference on Computational Complexity (2011)
San Jose, California USA
June 8, 2011 to June 11, 2011
ISSN: 1093-0159
ISBN: 978-0-7695-4411-3
pp: 300-308
We consider a system of linear constraints over any finite Abelian group G of the following form: l_i(x_1,...,x_n) = l_{i,1}x_1 + ... + l_{i,n}x_n in A_i for i=1,...,N and each A_i is subset of G and 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 of AND of ANY_{O(1)} of MOD-m for computing the MOD-q function, when m,q are co-prime. No super-polynomial 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).
lower bounds, boolean circuit complexity, modular gates, composite moduli, exponential sums

A. Chattopadhyay and S. Lovett, "Linear Systems over Finite Abelian Groups," 2011 IEEE 26th Annual Conference on Computational Complexity(CCC), San Jose, California USA, 2011, pp. 300-308.
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