Abstract
Maximum rank-distance (MRD) codes are extremal codes in the space of \(m\times n\) matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that \(2\leqslant m\leqslant n-2\).
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Acknowledgements
Yue Zhou would like to thank the hospitality of the University of Augsburg during his staying as a Fellow of the Alexander von Humboldt Foundation. This work is partially supported by the National Natural Science Foundation of China (Nos. 11401579, 11771451).
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Communicated by G. Lunardon.
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Schmidt, KU., Zhou, Y. On the number of inequivalent Gabidulin codes. Des. Codes Cryptogr. 86, 1973–1982 (2018). https://doi.org/10.1007/s10623-017-0433-6
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DOI: https://doi.org/10.1007/s10623-017-0433-6