Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decoding up to Hartmann-Tzeng and Roos bounds for rank codes

José Manuel Muñoz

TL;DR

The paper defines a unified family of rank-metric codes $C_{(\sigma,\mathbf{h},T)}$ that generalizes Gabidulin and skew cyclic codes by using left-kernel constructions tied to a defining set $T$ and a twist automorphism $\sigma$. It establishes Hartmann–Tzeng–like and Roos–like lower bounds on the rank distance and provides syndrome-based nearest-neighbor decoding algorithms built around multisequence skew-feedback shift-register synthesis, together with conditions guaranteeing decoding up to the bounds. The work also develops error-values/locators formalisms, key equations, and Gabidulin-inspired steps to solve the decoding problem, while analyzing decoding failures and practical criteria via Assumption $\text{Tnu-subset}$. Furthermore, it shows that subfield subcodes and interleaved codes allow unbounded code lengths and proves rank-equivalence between those approaches and the base skew-cyclic framework, enabling scalable, long-code constructions for applications like random linear network coding. Overall, the paper delivers bounds, decoding algorithms, and structural insights that unify rank-metric coding with skew-polynomial and defining-set perspectives, offering practical tools for robust rank-distance decoding at and beyond traditional limits.

Abstract

A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann-Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.

Decoding up to Hartmann-Tzeng and Roos bounds for rank codes

TL;DR

The paper defines a unified family of rank-metric codes that generalizes Gabidulin and skew cyclic codes by using left-kernel constructions tied to a defining set and a twist automorphism . It establishes Hartmann–Tzeng–like and Roos–like lower bounds on the rank distance and provides syndrome-based nearest-neighbor decoding algorithms built around multisequence skew-feedback shift-register synthesis, together with conditions guaranteeing decoding up to the bounds. The work also develops error-values/locators formalisms, key equations, and Gabidulin-inspired steps to solve the decoding problem, while analyzing decoding failures and practical criteria via Assumption . Furthermore, it shows that subfield subcodes and interleaved codes allow unbounded code lengths and proves rank-equivalence between those approaches and the base skew-cyclic framework, enabling scalable, long-code constructions for applications like random linear network coding. Overall, the paper delivers bounds, decoding algorithms, and structural insights that unify rank-metric coding with skew-polynomial and defining-set perspectives, offering practical tools for robust rank-distance decoding at and beyond traditional limits.

Abstract

A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann-Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.
Paper Structure (17 sections, 29 theorems, 74 equations, 3 algorithms)

This paper contains 17 sections, 29 theorems, 74 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Defining sets and lower bounds
  3. Skew polynomials and skew cyclic codes
  4. The multisequence skew-feedback shift-register synthesis problem
  5. Error values and error locators
  6. The implicit SFSR synthesis problems in the error-correcting problem
  7. On the nature of Assumption \ref{['assumption:Tnusubset']}
  8. Identifying and handling decoding failures
  9. Gabidulin's algorithm
  10. An example
  11. Applying the bounds
  12. Applying the assumptions
  13. Running the algorithms
  14. Forcing decoding failures
  15. Increasing the length of the codes
  16. ...and 2 more sections

Key Result

Lemma 4

All codes of length $n$ satisfying Definition defn:ccode for the same $\sigma, T$ and such that their respective ${\normalfont \textbf{h}} \in F^n$ span the same subspace of $F$ are $F^\sigma$-linearly rank equivalent.

Theorems & Definitions (87)

  • Definition 1
  • Remark 2
  • Definition 3
  • Lemma 4
  • proof
  • Corollary 5
  • Definition 6
  • Theorem 7: Hartmann--Tzeng bound
  • proof
  • Theorem 8: Roos bound
  • ...and 77 more