Several classes of optimal Ferrers diagram rank-metric codes

TL;DR

This paper introduces four new constructions for Ferrers diagram rank-metric codes, unifying and extending existing methods to produce optimal codes with various parameters and representations.

Contribution

The paper presents four novel constructions for FDRM codes, including a unifying approach using restricted Gabidulin codes and two representations of finite field elements.

Findings

All four constructions produce optimal FDRM codes.

The methods encompass various Ferrers diagram shapes and parameters.

The second construction unifies many known FDRM code constructions.

Abstract

Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field $\mathbb F_{q^m}$ (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and parameters.