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A class of constacyclic codes are generalized Reed–Solomon codes

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Abstract

Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed–Solomon (GRS) codes. The square \(\mathcal {C}^{2}\) of a linear code \(\mathcal {C}\) is the linear code spanned by the component-wise products of every pair of codewords in \(\mathcal {C}\). For an MDS code \(\mathcal {C}\), it is convenient to determine whether \(\mathcal {C}\) is a GRS code by determining the dimension of \(\mathcal {C}^{2}\). In this paper, we investigate under what conditions that MDS constacyclic codes are GRS. For this purpose, we first study the square of constacyclic codes. Then, we give a sufficient condition that a constacyclic code is GRS. In particular, we provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS.

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Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant Nos. 12271199, 12171191 and The Fundamental Research Funds for the Central Universities 30106220482. We sincerely thank the reviewer for their valuable comments which have improved the presentation of this paper.

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  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Hongwei Liu & Shengwei Liu

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  1. Hongwei Liu
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Correspondence to Shengwei Liu.

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Communicated by J. Bierbrauer.

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Liu, H., Liu, S. A class of constacyclic codes are generalized Reed–Solomon codes. Des. Codes Cryptogr. 91, 4143–4151 (2023). https://doi.org/10.1007/s10623-023-01294-6

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