Properties of Rank Metric Codes

TL;DR

This paper explores fundamental properties of rank metric codes, including their packing, covering, and weight distribution, establishing new bounds and identities that deepen understanding of their structure and relationships.

Contribution

It introduces new bounds on rank metric code parameters, analyzes their asymptotic properties, and derives identities linking code and dual code weight distributions, including a novel MacWilliams-type identity.

Findings

Derived bounds on rank metric code parameters

Analyzed asymptotic packing and covering properties

Established a new identity relating rank weight distributions

Abstract

This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their parameters, and investigate their asymptotic covering properties. Finally, we establish several identities that relate the rank weight distribution of a linear code to that of its dual code. One of our identities is the counterpart of the MacWilliams identity for the Hamming metric, and it has a different form from the identity by Delsarte.