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2011 26th Annual IEEE Conference on Computational Complexity
Noisy Interpolation of Sparse Polynomials, and Applications
San Jose, California USA
June 08June 11
ISBN: 9780769544113
ASCII Text  x  
Shubhangi Saraf, Sergey Yekhanin, "Noisy Interpolation of Sparse Polynomials, and Applications," 2012 IEEE 27th Conference on Computational Complexity, pp. 8692, 2011 26th Annual IEEE Conference on Computational Complexity, 2011.  
BibTex  x  
@article{ 10.1109/CCC.2011.38, author = {Shubhangi Saraf and Sergey Yekhanin}, title = {Noisy Interpolation of Sparse Polynomials, and Applications}, journal ={2012 IEEE 27th Conference on Computational Complexity}, volume = {0}, year = {2011}, issn = {10930159}, pages = {8692}, doi = {http://doi.ieeecomputersociety.org/10.1109/CCC.2011.38}, 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  Noisy Interpolation of Sparse Polynomials, and Applications SN  10930159 SP86 EP92 A1  Shubhangi Saraf, A1  Sergey Yekhanin, PY  2011 KW  Sparse polynomials KW  interpolation KW  locally decodable codes KW  matrix rigidity VL  0 JA  2012 IEEE 27th Conference on Computational Complexity ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.38
Let f in F_q[x] be a polynomial of degree d _ q=2: It is wellknown that f can be uniquely recovered from its values at some 2d points even after some small fraction of the values are corrupted. In this paper we establish a similar result for sparse polynomials. We show that a ksparse polynomial f 2 Fq[x] of degree d _ q=2 can be recovered from its values at O(k) randomly chosen points, even if a small fraction of the values of f are adversarially corrupted. Our proof relies on an iterative technique for analyzing the rank of a random minor of a matrix.We use the same technique to establish a collection of other results. Specifically, _ We show that restricting any linear [n; k; _n]q code to a randomly chosen set of O(k) coordinates with high probability yields an asymptotically good code. _ We improve the state of the art in locally decodable codes, showing that similarly to Reed Muller codes matching vector codes require only a constant increase in query complexity in order to tolerate a constant fraction of errors. This result yields a moderate reduction in the query complexity of the currently best known codes. _ We improve the state of the art in constructions of explicit rigid matrices. For any prime power q and integers n and d we construct an explicit matrix M with exp(d) _ n rows and n columns such that the rank of M stays above n=2 even if every row of M is arbitrarily altered in up to d coordinates. Earlier, such constructions were available only for q = O(1) or q = (n):
Index Terms:
Sparse polynomials, interpolation, locally decodable codes, matrix rigidity
Citation:
Shubhangi Saraf, Sergey Yekhanin, "Noisy Interpolation of Sparse Polynomials, and Applications," ccc, pp.8692, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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