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 $v \in \mathbb{F}_2^n$ and a linear space $L\subseteq \mathbb{F}_2^n$ 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 /logn ). 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 /logn )-approximation algorithm; An n O (s ) time O (k log(s ) n / logn )-approximation algorithm, where log(s ) n stands for s iterations of log, e.g., log(2) n = loglogn ; An $n^{O(\log^* n)}$ time O (k /logn )-approximation algorithm. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space $L\subseteq \mathbb{F}_2^n$ of dimension k RPP asks to find a point $v\in \mathbb{F}_2^n$ 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 ***(n logk / 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.