Generalized rank weights : a duality statement

TL;DR

This paper establishes a duality relationship for generalized rank weights of linear codes over finite field extensions, paralleling Wei's results for Hamming weights, and characterizes cases of equality in the rank weight Singleton bound.

Contribution

It introduces a duality statement for generalized rank weights and characterizes the equality case of the generalized rank weight Singleton bound.

Findings

Derived a duality statement for generalized rank weights.

Characterized the equality case of the generalized rank weight Singleton bound.

Extended Wei's characterization from Hamming to rank weights.

Abstract

We consider linear codes over some fixed finite field extension over an arbitrary finite field. Gabidulin introduced rank metric codes, by endowing linear codes over the extension field with a rank weight over the base field and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of wiretap II codes in terms of generalized Hamming weights by Wei, Kurihara et al. defined some generalized rank weights and showed their relevance for secure network coding. In this paper, we derive a statement for generalized rank weights of the dual code, completely analogous to Wei's one for generalized Hamming weights and we characterize the equality case of the r-generalized Singleton bound for the generalized rank weights, in terms of the rank weight of the dual code.