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2011 26th Annual IEEE Conference on Computational Complexity
Linear Systems over Finite Abelian Groups
San Jose, California USA
June 08June 11
ISBN: 9780769544113
ASCII Text  x  
Arkadev Chattopadhyay, Shachar Lovett, "Linear Systems over Finite Abelian Groups," 2012 IEEE 27th Conference on Computational Complexity, pp. 300308, 2011 26th Annual IEEE Conference on Computational Complexity, 2011.  
BibTex  x  
@article{ 10.1109/CCC.2011.25, author = {Arkadev Chattopadhyay and Shachar Lovett}, title = {Linear Systems over Finite Abelian Groups}, journal ={2012 IEEE 27th Conference on Computational Complexity}, volume = {0}, year = {2011}, issn = {10930159}, pages = {300308}, doi = {http://doi.ieeecomputersociety.org/10.1109/CCC.2011.25}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2012 IEEE 27th Conference on Computational Complexity TI  Linear Systems over Finite Abelian Groups SN  10930159 SP300 EP308 A1  Arkadev Chattopadhyay, A1  Shachar Lovett, PY  2011 KW  lower bounds KW  boolean circuit complexity KW  modular gates KW  composite moduli KW  exponential sums VL  0 JA  2012 IEEE 27th Conference on Computational Complexity ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.25
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 MODq boolean function, when the order of G and q are coprime 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 depthfour circuits of the form MAJ of AND of ANY_{O(1)} of MODm for computing the MODq function, when m,q are coprime. 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).
Index Terms:
lower bounds, boolean circuit complexity, modular gates, composite moduli, exponential sums
Citation:
Arkadev Chattopadhyay, Shachar Lovett, "Linear Systems over Finite Abelian Groups," ccc, pp.300308, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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