Self-duality of generalized twisted Gabidulin codes

Abstract

Self-duality of Gabidulin codes was investigated in [10] and the authors provided an if and only if condition for a Gabidulin code to be equivalent to a self-dual maximum rank distance (MRD) code. In this paper, we investigate the analog problem for generalized twisted Gabidulin codes (a larger family of linear MRD codes including the family of Gabidulin codes). We observe that the condition presented in [10] still holds for generalized Gabidulin codes (an intermediate family between Gabidulin codes and generalized twisted Gabidulin codes). However, beyond the family of generalized Gabidulin codes we observe that some additional conditions are required depending on the additional parameters. Our tools are similar to those in [10] but we also use linearized polynomials, which leads to further tools and direct proofs.

Hermitian Rank Metric Codes and Duality

“…Rank metric codes were introduced by Delsarte (1978) as a q-analogue of coding theory [13]. Due to their applications in cryptography and in network error correction ( [30] [31]), there is a great interest in studying their general properties and their connections with other topics [1], [4], [9]- [11], [24], [26], [27], [29].…”

“…Additive codes and self-duality are also considered in the ambient space of matrices endowed with the rank metric (see, for example, [22], [25], [26]). They have potential applications not only in network coding, combinatorics and cryptography but also in code-based cryptography (and hence post-quantum cryptography).…”