Minimun distance decoding algorithms for linear codes

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.

Research done while at Faculty of Math, and Computing Science, Technical University of Eindhoven, Eindhoven, The Netherlands.