Generalized Twisted Gabidulin Codes

Abstract

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 $m⩽n$. 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.

One missing case (n=4) has been included in the appendix. Typos are corrected, Journal of Combinatorial Theory, Series A, 2018

References in corpus (4)

• Algebraic structures of MRD Codes
• Properties of codes in rank metric
• A new family of MRD codes in ${\mathbb{F}}_q^{2n\times 2n}$ with right and middle nuclei ${\mathbb{F}}_{q^n}$
• Properties of Codes with the Rank Metric

Cited by in corpus (24)

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• Linear sets and MRD-codes arising from a class of scattered linearized polynomials
• MRD codes with maximum idealizers
• A Carlitz type result for linearized polynomials
• Identifiers for MRD-codes
• Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants
• Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes
• Nuclei and automorphism groups of generalized twisted Gabidulin codes
• Towards the full classification of exceptional scattered polynomials
• Automorphism groups and new constructions of maximum additive rank metric codes with restrictions
• On a conjecture about maximum scattered subspaces of ${\mathbb{F}}_{q^6}\times {\mathbb{F}}_{q^6}$
• Exceptional Scattered Polynomials
• Asymptotics of Moore exponent sets
• Decoding High-Order Interleaved Rank-Metric Codes
• Invariants and Inequivalence of Linear Rank-Metric Codes
• A condition for scattered linearized polynomials involving Dickson matrices
• A note on rank-metric codes
• ${\mathbb{F}}_{q^n}$-linear rank distance codes and their distinguishers
• A remark on ${\mathbb{F}}_{q^n}$-Linear MRD codes
• A geometric invariant of linear rank-metric codes
• On the Sparseness of Certain MRD Codes
• Bilinear Complexity of 3-Tensors Linked to Coding Theory