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

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Abstract

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.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 12071321).

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Authors and Affiliations

  1. College of Mathematical Science, Sichuan Normal University, Chengdu, 610066, Sichuan, China

    Boyi He & Qunying Liao

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  1. Boyi He
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  2. Qunying Liao
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Correspondence to Qunying Liao.

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Communicated by J. Jedwab

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Supported by National Natural Science Foundation of China (Grant No. 12071321)

Appendix: MAGMA program

Appendix: MAGMA program

figure a

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He, B., Liao, Q. The properties and the error-correcting pair for lengthened GRS codes. Des. Codes Cryptogr. 92, 211–225 (2024). https://doi.org/10.1007/s10623-023-01304-7

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  • DOI: https://doi.org/10.1007/s10623-023-01304-7

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