Deterministic Approximation Algorithms for the Nearest Codeword Problem

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v ∈ Fn 2 and a linear space L ⊆ Fn 2 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) ...

• Type: Conference
• Year: 2009
• Where: APPROX
• Authors: Noga Alon, Rina Panigrahy, Sergey Yekhanin
• Keywords: Algorithm, Algorithms, APPROX 2009, Log N)-approximation Algorithm, Nearest Codeword Problem