Rank error-correcting pairs
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Error-correcting pairs were introduced as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some of which preserve symbolic products of linearized polynomials after evaluation and some of which coincide with usual products of matrices. Then we define rank error-correcting pairs for codes that are linear over the extension field and for codes that are linear over the base field, and relate both types. Bounds on the minimum rank distance of codes and MRD conditions are given. Finally we show that some well-known families of rank-metric codes admit rank error-correcting pairs, and show that the given algorithm generalizes the classical algorithm using error-correcting pairs for the Hamming metric.
KeywordsDecoding Error-correcting pairs Linearized polynomials Rank metric Vector products
Mathematics Subject Classification15B33 94B35 94B65
Rank error-correcting pairs of type II have been obtained in the case \( m = n \) independently by Alain Couvreur (About error correcting pairs. Personal communication, September 15, 2015). The authors wish to thank him for this communication and for useful discussions and comments. The authors wish to thank the anonymous reviewers for their helpful comments and also gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF-4002-00367). This paper was started during the visit of the second author to Aalborg University, Denmark, which was supported by the previous grant.
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