Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Completely decomposable rank-metric codes

Paolo Santonastaso

Abstract

In this paper, we investigate completely decomposable rank-metric codes, i.e. rank-metric codes that are the direct sum of 1-dimensional maximum rank distance codes. We study the weight distribution of such codes, characterizing codewords with certain rank weights. Additionally, we obtain classification results for codes with the largest number of minimum weight codewords within the class of completely decomposable codes.

Completely decomposable rank-metric codes

Abstract

In this paper, we investigate completely decomposable rank-metric codes, i.e. rank-metric codes that are the direct sum of 1-dimensional maximum rank distance codes. We study the weight distribution of such codes, characterizing codewords with certain rank weights. Additionally, we obtain classification results for codes with the largest number of minimum weight codewords within the class of completely decomposable codes.
Paper Structure (9 sections, 28 theorems, 125 equations)

This paper contains 9 sections, 28 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual of a subspace
  4. Product of subspaces
  5. Rank-metric codes
  6. The geometry of rank-metric codes
  7. Direct sum of 1-dimensional MRD codes
  8. On the weight distribution of completely decomposable codes
  9. Tightness of the bounds on minimum weight codewords

Key Result

Proposition 2.1

Let $U$ be an ${\mathbb F}_{q}$-subspace of $V$ and $W$ be an $\mathbb{F}_{q^m}$-subspace of $V$. Then

Theorems & Definitions (61)

  • Proposition 2.1: see polverino2010linear
  • Proposition 2.2: see napolitano2023classifications
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5: see hou2002generalization and bachoc2018revisiting
  • Theorem 2.6: see bachoc2017analogue and bachoc2017analogue
  • Lemma 2.7
  • proof
  • ...and 51 more