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On equivalence of maximum additive symmetric rank-distance codes

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Abstract

Let \({\mathcal {S}}_m({\mathbb {F}})\) denote the set of symmetric matrices over a finite field \({\mathbb {F}}\). Let \({\mathcal {C}}\) be an additive subset of \({\mathcal {S}}_m({\mathbb {F}})\) which has minimum rank-distance d and \(|{\mathcal {C}}|\) meets the upper bound. We call \({\mathcal {C}}\) a maximum additive d-code. In particular, when \(d=m\), \({\mathcal {C}}\) is equivalent to a symplectic semifield spreadset. For a given invertible matrix P, \(S_0\in {\mathcal {S}}_m({\mathbb {F}})\) and nonzero \(a\in {\mathbb {F}}\), it is easy to see that the map \(M \mapsto aP^T M^\sigma P + S_0\) defines an isometry on \({\mathcal {S}}_m({\mathbb {F}})\) with respect to the rank-metric. However, by examples of symplectic semifields, it is already known that there exist other maps preserving the rank-distance of maximum additive m-codes. In this paper, we prove that this can never happen when \(d<m\). We also present a new construction of maximum additive 2-codes in \({\mathcal {S}}_m({\mathbb {F}})\) for odd m which are not t-designs.

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Acknowledgements

The author would like to thank Kai-Uwe Schmidt for his valuable comments and the anonymous referee for pointing out reference [14]. This work is supported by the National Natural Science Foundation of China under Grant No. 11771451 and Natural Science Foundation of Hunan Province under Grant No. 2019JJ30030.

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Correspondence to Yue Zhou.

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Communicated by K.-U. Schmidt.

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Zhou, Y. On equivalence of maximum additive symmetric rank-distance codes. Des. Codes Cryptogr. 88, 841–850 (2020). https://doi.org/10.1007/s10623-020-00716-z

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