Generalized Twisted Gabidulin Codes

Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$. Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then $\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1<d<m-1$. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.Comment: One missing case (n=4) has been included in the appendix. Typos are corrected, Journal of Combinatorial Theory, Series A, 201

A decoding algorithm for Twisted Gabidulin codes

In this work, we modify the decoding algorithm for subspace codes by Koetter and Kschischang to get a decoding algorithm for (generalized) twisted Gabidulin codes. The decoding algorithm we present applies to cases where the code is linear over the base field $\mathbb{F}_q$ but not linear over $\mathbb{F}_{q^n}$.Comment: This paper was submitted to ISIT 201

Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. The same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we derive upper and lower bounds on the number of equivalence classes of Gabidulin codes and twisted Gabidulin codes, improving a result of Schmidt and Zhou for a wider range of parameters. In the end we use the aforementioned sequences to determine a characterization result for Gabidulin codes.Comment: 37 pages, 1 figure, 3 tables, extended version of arXiv:1905.1132

Puncturing maximum rank distance codes

We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models, we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes, and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey (Adv Math Commun 10:475–488, 2016)

On interpolation-based decoding of a class of maximum rank distance codes

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio

Puncturing maximum rank distance codes

We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models, we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes, and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey (Adv Math Commun 10:475–488, 2016)