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 ∈ double-struck F ₂ ⁿ and a linear space L ⊆ double-struck 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/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/logn)-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 ⊆ double-struck F ₂ ⁿ of dimension k RPP asks to find a point v ∈ double-struck 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(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.