Rank-metric LCD codes

TL;DR

This paper explores the properties of rank-metric LCD codes, establishing relationships between different classes and constructing new Gabidulin LCD MRD codes using special bases, with conditions for optimal anticodes.

Contribution

It introduces new classes of Gabidulin LCD MRD codes using self-dual bases and provides conditions for Delsarte optimal anticodes to be LCD codes.

Findings

Constructed two classes of Gabidulin LCD MRD codes.

Established relationship between Delsarte and Gabidulin LCD codes.

Provided sufficient conditions for Delsarte optimal anticodes to be LCD.

Abstract

In this paper, we investigate the rank-metric codes which are proposed by Delsarte and Gabidulin to be complementary dual codes. We point out the relationship between Delsarte complementary dual codes and Gabidulin complementary dual codes. In finite field $\mathbb{F}_{q}^{m}$, we construct two classes of Gabidulin LCD MRD codes by self-dual basis (or almost self-dual basis) of $\mathbb{F}_{q}^{m}$ over $\mathbb{F}_{q}$. Under a suitable condition, we determine a sufficient condition for Delsarte optimal anticodes to be LCD codes over $\mathbb{F}_{q}$.