The properties and the error-correcting pair for lengthened GRS codes

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has goo...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2024, Vol.92 (1), p.211-225
Main Authors: He, Boyi, Liao, Qunying
Format: Article
Language:English
Subjects:
Algebra
Coding and Information Theory
Computer Science
Cryptology
Decoding
Discrete Mathematics in Computer Science
Error correction
Reed-Solomon codes
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Summary:The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has good algebraic structure and many excellent properties, thus we extend the error-correcting pair to the case for lengthened generalized Reed-Solomon codes. Firstly, we give some sufficient conditions for which an LGRS code is non-GRS, and a necessary and sufficient condition for an LGRS code to be MDS or AMDS, respectively. And then, we constructively determine the existence of the error-correcting pair for lengthened generalized Reed-Solomon codes and give several examples to support our main results.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-023-01304-7