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.
Similar content being viewed by others
References
Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217(1), 282–304 (2008).
Csajbók B., Marino G., Polverino O., Zullo F.: Maximum scattered linear sets and mrd-codes. J. Algebraic Comb. 46(3), 517–531 (2017).
Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory A 25(3), 226–241 (1978).
Gabidulin E.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21, 3–16 (1985).
Johnson N.L., Jha V., Biliotti M.: Handbook of Finite Translation Planes, vol. 289. Pure and Applied MathematicsChapman & Hall/CRC, Boca Raton, FL (2007).
Lavrauw M., Polverino O.: Finite semifields. In: Storme L., De Beule J. (eds.) Current Research Topics in Galois Geometry, pp. 131–160. NOVA Academic Publishers, New York (2011). chapter 6.
Liebhold D., Nebe G.: Automorphism groups of Gabidulin-like codes. Arch. Math. 107(4), 355–366 (2016).
Lunardon G.: MRD-codes and linear sets. J. Comb. Theory A 149, 1–20 (2017).
Lunardon G., Trombetti R., Zhou Y.: On kernels and nuclei of rank metric codes. J. Algebraic Comb. 46(2), 313–340 (2017).
Schmidt K.-U.: Symmetric bilinear forms over finite fields of even characteristic. J. Comb. Theory A 117(8), 1011–1026 (2010).
Schmidt K.-U.: Symmetric bilinear forms over finite fields with applications to coding theory. J. Algebraic Comb. 42(2), 635–670 (2015).
Schmidt K.-U.: Quadratic and symmetric bilinear forms over finite fields and their association schemes. to appear in Algebraic Combinatorics. arXiv:1803.04274
Schmidt K.-U., Zhou Y.: On the number of inequivalent Gabidulin codes. Des. Codes Cryptogr. 86(9), 1973–1982 (2018).
Schmidt M.: Rank metric codes. Master’s thesis, University of Bayreuth, Germany (2016)
Sheekey J.: MRD codes: constructions and connections. arXiv:1904.05813 [math] 4 (2019)
Wan Z.: Geometry of Matrices. World Scientific, Singapore (1996).
Zhou Y., Pott A.: A new family of semifields with 2 parameters. Adv. Math. 234, 43–60 (2013).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K.-U. Schmidt.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-020-00716-z