Constructions for optimal Ferrers diagram rank-metric codes

TL;DR

This paper introduces new constructions for optimal Ferrers diagram rank-metric codes, generalizing previous methods and providing explicit systematic MRD code constructions to enhance subspace code design.

Contribution

It generalizes existing constructions for FDRM codes and introduces proper combinations of Ferrers diagrams for improved code optimality.

Findings

Explicit systematic MRD code construction presented

Conditions for optimal FDRM codes established

Generalization of combining constructions for Ferrers diagrams

Abstract

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criteria for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring subcodes of Gabidulin codes, if each of the rightmost $\delta-1$ columns in Ferrers diagram $\cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $\cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.