Deterministic Approximation Algorithms for the Nearest Codeword Problem
Abstract
The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of errorcorrecting 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 wellknown that the nearest codeword problem is NPhard. 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 rfar 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.
Keywords
Approximation Algorithm Linear Space Approximation Ratio Linear Code Deterministic AlgorithmPreview
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