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 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 n^O(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 n^{O(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.