# Minimun distance decoding algorithms for linear codes

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## Abstract

Some known general (i.e., applicable to all linear codes) algorithms of minimum distance decoding use a certain set of codewords to successively improve the current decision. Important examples in this class are the minimal-vectors decoding [14] and zero-neighbours decoding [16]. These two methods have some common features that enable us to introduce a general *gradient-like decoding algorithm*. We formulate key properties of this decoding, which allows us to analyze known algorithms in this class in a simple and unified manner. Further, we show that under certain conditions, gradient-like algorithms must examine all zero neighbours, and therefore, the size of this set provides a *lower* bound on the complexity of algorithms in this class. This implies that general asymptotic improvements of the zero-neighbours decoding algorithm in the class of gradient-like methods are impossible.

The second group of methods, *information set decoding*, is better studied and yields algorithms with best known asymptotic complexity. All known algorithms in this group can be viewed as modifications of covering set decoding. Following [4], we introduce an algorithm that for long codes ensures the maximum likelihood performance and has the smallest known asymptotic complexity for any code rate *R, 0<R<1*.

## Keywords

Linear Code Code Rate Decode Algorithm Voronoi Region Decode Method## Preview

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