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$r$-Minimal Codes with Respect to Rank Metric

Yang Xu, Haibin Kan, Guangyue Han

TL;DR

This work introduces $r$-minimal codes as a broad generalization of minimal codes across Hamming, rank, and sum-rank metrics, framed within a lattice-based, division-ring module setting. The authors develop a unifying sigma-map framework that specializes to standard metrics, establish a general Singleton-type bound, and prove existence results; they then connect $r$-minimal rank-metric codes to linear cutting $r$-blocking sets and generalized rank weights, deriving tight bounds on minimal length via evasive subspaces. A key contribution is the equivalence between $r$-minimality and cutting $r$-blocking sets, enabling dual-weight characterizations and a detailed study of minimal lengths, including explicit results such as $oldsymbol{ u}_{bE/bF}(k,1)$ bounds and exact values for small $m$ (notably $m=3$ gives $oldsymbol{ u}_{bE/bF}(k,1)=2k$). These results advance the understanding of the geometry–coding theory interplay in the rank and sum-rank settings and supply tools for constructing and bounding $r$-minimal codes in both finite and infinite alphabets.

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.

$r$-Minimal Codes with Respect to Rank Metric

TL;DR

This work introduces -minimal codes as a broad generalization of minimal codes across Hamming, rank, and sum-rank metrics, framed within a lattice-based, division-ring module setting. The authors develop a unifying sigma-map framework that specializes to standard metrics, establish a general Singleton-type bound, and prove existence results; they then connect -minimal rank-metric codes to linear cutting -blocking sets and generalized rank weights, deriving tight bounds on minimal length via evasive subspaces. A key contribution is the equivalence between -minimality and cutting -blocking sets, enabling dual-weight characterizations and a detailed study of minimal lengths, including explicit results such as bounds and exact values for small (notably gives ). These results advance the understanding of the geometry–coding theory interplay in the rank and sum-rank settings and supply tools for constructing and bounding -minimal codes in both finite and infinite alphabets.

Abstract

In this paper, we propose and study -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 -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 -minimal codes, derive a general singleton bound, and give existence results for -minimal codes by using combinatorial arguments. We then consider -minimal rank metric codes over a field extension of degree , where can be infinite. We characterize these codes in terms of cutting -blocking sets, generalized rank weights of the codes and those of the dual codes, and classify codes whose -dimensional subcodes have constant rank support weight. Next, with the help of the evasiveness property of cutting -blocking sets and some upper bounds for the dimensions of evasive subspaces, we derive several lower and upper bounds for the minimal length of -minimal codes. Furthermore, when 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 , then for any , the minimal length of -dimensional minimal codes is equal to . To the best of our knowledge, when , there was no known explicit formula for the minimal length of -dimensional minimal codes for arbitrary in the literature.
Paper Structure (17 sections, 40 theorems, 64 equations)

This paper contains 17 sections, 40 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rank metric codes
  4. Evasive subspaces and cutting $r$-blocking sets
  5. Some preliminaries for $q$-binomial coefficient
  6. General $r$-minimal codes supported on lattices
  7. Examples
  8. $\sigma$-Minimal subcodes and $(\sigma,r)$-minimal codes
  9. Existence results for $(\sigma,r)$-minimal codes
  10. Cutting $r$-blocking sets and $r$-minimal rank metric codes
  11. An extension of Lemma 2.2
  12. Necessary and sufficient conditions for linear cutting $r$-blocking sets
  13. Necessary and sufficient conditions for $r$-minimal codes
  14. Minimal length of $r$-minimal codes
  15. Evasive subspaces
  16. ...and 2 more sections

Key Result

Lemma 2.1

(1) For $A\subseteq\mathbb{E}^{n}$, we have $A^{\bot}\cap\mathbb{F}^{n}=\chi(A)^{\bot}\cap\mathbb{F}^{n}$ and $\mathrm{wt}\,(A)=n-\dim_{\mathbb{F}}(A^{\bot}\cap\mathbb{F}^{n})$. (2) Let $k\in\mathbb{Z}^{+}$, and let $G\in\mathrm{Mat}\,_{k,n}(\mathbb{E})$, $C=\langle\mathrm{row}\,(G)\rangle_{\mathb

Theorems & Definitions (100)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Corollary 2.1
  • proof
  • Definition 2.1
  • Remark 2.2
  • Definition 2.2
  • Lemma 2.2
  • Remark 2.3
  • ...and 90 more