$r$-Minimal Codes with Respect to Rank Metric

TL;DR

This paper introduces and studies $r$-minimal codes, extending the concept of minimal codes to various metrics and settings, providing bounds, characterizations, and explicit formulas for their minimal lengths.

Contribution

It generalizes minimal codes to $r$-minimal codes over division rings and fields, characterizes their structure, and derives bounds and explicit formulas for minimal lengths.

Findings

Characterization of $r$-minimal codes in general settings

Derivation of bounds for minimal length of $r$-minimal codes

Explicit formula for minimal length when $m=3$

Abstract

In this paper, we propose and study $r$-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose $r$-minimal codes in a general setting where the ambient space is a finite dimensional left module over a division ring and is supported on a lattice. We characterize minimal subcodes and $r$-minimal codes, derive a general singleton bound, and give existence results for $r$-minimal codes by using combinatorial arguments. We then consider $r$-minimal rank metric codes over a field extension $\mathbb{E}/\mathbb{F}$ of degree $m$, where $\mathbb{E}$ can be infinite. We characterize these codes in terms of cutting $r$-blocking sets, generalized rank weights of the codes and those of the dual codes, and classify codes whose $r$-dimensional subcodes have constant rank support weight. Next, with the help of the evasiveness property of cutting $r$-blocking sets and some upper bounds for the dimensions of evasive subspaces, we derive several lower and upper bounds for the minimal length of $r$-minimal codes. Furthermore, when $\mathbb{E}$ is finite, we establish a general upper bound which generalizes and improves the counterpart for minimal codes in the literature. As a corollary, we show that if $m=3$, then for any $k\geqslant2$, the minimal length of $k$-dimensional minimal codes is equal to $2k$. To the best of our knowledge, when $m\geqslant3$, there was no known explicit formula for the minimal length of $k$-dimensional minimal codes for arbitrary $k$ in the literature.