On the number of inequivalent MRD codes

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 $2\le m\le n-2$.

[1]  Katherine Morrison,et al.  Equivalence for Rank-Metric and Matrix Codes and Automorphism Groups of Gabidulin Codes , 2013, IEEE Transactions on Information Theory.

[2]  Rod Gow,et al.  Galois theory and linear algebra , 2009 .

[3]  Giuseppe Marino,et al.  Non-linear maximum rank distance codes , 2016, Des. Codes Cryptogr..

[4]  W. Kantor,et al.  Orthogonal dual hyperovals, symplectic spreads, and orthogonal spreads , 2013, 1303.4073.

[5]  Rocco Trombetti,et al.  Nuclei and automorphism groups of generalized twisted Gabidulin codes , 2016, Linear Algebra and its Applications.

[6]  John Sheekey,et al.  A new family of linear maximum rank distance codes , 2015, Adv. Math. Commun..

[7]  Anna-Lena Horlemann-Trautmann,et al.  New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions , 2015, Adv. Math. Commun..

[8]  Guglielmo Lunardon,et al.  MRD-codes and linear sets , 2017, J. Comb. Theory, Ser. A.

[9]  Rocco Trombetti,et al.  Generalized Twisted Gabidulin Codes , 2015, J. Comb. Theory A.

[10]  B. Huppert Endliche Gruppen I , 1967 .

[11]  Ulrich Dempwolff,et al.  Dimensional dual hyperovals and APN functions with translation groups , 2014 .

[12]  Alessandro Neri,et al.  On the genericity of maximum rank distance and Gabidulin codes , 2016, Des. Codes Cryptogr..

[13]  Gabriele Nebe,et al.  Automorphism groups of Gabidulin-like codes , 2016, ArXiv.

[14]  Ernst M. Gabidulin,et al.  The new construction of rank codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[15]  Rocco Trombetti,et al.  On kernels and nuclei of rank metric codes , 2016, ArXiv.

[16]  Giuseppe Marino,et al.  Maximum scattered linear sets and MRD-codes , 2017, 1701.06831.

[17]  Alfred Wassermann,et al.  Algebraic structures of MRD codes , 2015, Adv. Math. Commun..