TY - JOUR

T1 - Deterministic Approximation Algorithms for the Nearest Codeword Problem

AU - Alon, Noga

AU - Panigrahy, Rina

AU - Yekhanin, Sergey

N1 - Publisher Copyright:
© 2009 Algebraic Methods in Computational Complexity. All Rights Reserved.

PY - 2010

Y1 - 2010

N2 - The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v Fn2 and a linear space L Fn2 of dimension k NCP asks to find a point l L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best efficient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c, and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work. Specifically, we obtain: – A polynomial time O(n/ log n)-approximation algorithm; – An nO(s) time O(k log(s) n/ log n)-approximation algorithm, where log(s) n stands for s iterations of log, e.g., log(2) n = log log n; – An nO(log* n) time O(k/ log n)-approximation algorithm. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L Fn2 of dimension k RPP asks to find a point v Fn2 that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Ω(n log k/k) for all k ≤ n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.

AB - The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v Fn2 and a linear space L Fn2 of dimension k NCP asks to find a point l L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best efficient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c, and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work. Specifically, we obtain: – A polynomial time O(n/ log n)-approximation algorithm; – An nO(s) time O(k log(s) n/ log n)-approximation algorithm, where log(s) n stands for s iterations of log, e.g., log(2) n = log log n; – An nO(log* n) time O(k/ log n)-approximation algorithm. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L Fn2 of dimension k RPP asks to find a point v Fn2 that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Ω(n log k/k) for all k ≤ n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.

UR - http://www.scopus.com/inward/record.url?scp=85173869985&partnerID=8YFLogxK

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AN - SCOPUS:85173869985

SN - 1862-4405

VL - 9421

JO - Dagstuhl Seminar Proceedings

JF - Dagstuhl Seminar Proceedings

T2 - Algebraic Methods in Computational Complexity 2009

Y2 - 11 October 2009 through 16 October 2009

ER -