Deterministic Approximation Algorithms for the Nearest Codeword Problem

Abstract

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 Ω(nlogk / 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.