Maximum weight codewords of a linear rank metric code

TL;DR

This paper studies the number of maximum weight codewords in linear rank metric codes over finite fields, aiming to characterize their distribution and extremal cases.

Contribution

It investigates the enumeration of maximum weight codewords in linear rank metric codes and characterizes codes with extremal counts.

Findings

Derived formulas for the number of maximum weight codewords.

Characterized codes with minimal and maximal counts of such codewords.

Provided insights into the structure of rank metric codes.

Abstract

Let $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear non-degenerate rank metric code with dimension $k$. In this paper we investigate the problem of determining the number $M(\mathcal{C})$ of codewords in $\mathcal{C}$ with maximum weight, that is $\min\{m,n\}$, and to characterize those with the maximum and the minimum values of $M(\mathcal{C})$.