Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decoding Trombetti-Zhou codes: a new syndrome-based decoding approach

Chunlei Li, Angelica Piccirillo, Olga Polverino, Ferdinando Zullo

TL;DR

The paper develops a syndrome-based decoder for Trombetti–Zhou (TZ) rank-metric codes, which are $ ext{F}_{q^n}$-linear over $ ext{F}_{q^{2n}}$ and not fully $ ext{F}_{q^{2n}}$-linear. It introduces $ ext{F}_{q^n}$-generator and parity-check matrices built from a trace almost dual basis, enabling a syndrome-based decoding framework that, for error rank $t< rac{d-1}{2}$, reduces to decoding Gabidulin codes of dimension $1$ larger, while for the critical case $t= rac{d-1}{2}$ it reduces to a structured rank-determination problem. The method constructs an error-span polynomial $oldsymbol{ ext{Λ}}(x)$, recovers an error decomposition $oldsymbol{e}=oldsymbol{a} oldsymbol{B}$, and reconstructs the original codeword; a detailed complexity analysis shows a cubic complexity in the half-length over $ ext{F}_{q^{2n}}$. The work thus provides a natural decoder for TZ codes that can exploit erasures/deviations and advances the understanding of subfield-linear rank-metric codes with practical decoding guarantees. Open problems include extending beyond the strict unique-decoding-radius constraint and improving efficiency relative to interpolation-based decoders.

Abstract

In 2019, Trombetti and Zhou introduced a new family of $\mathbb{F}_{q^n}$-linear Maximum Rank Distance (MRD) codes over $\mathbb{F}_{q^{2n}}$. For such codes we propose a new syndrome-based decoding algorithm. It is well known that a syndrome-based decoding approach relies heavily on a parity-check matrix of a linear code. Nonetheless, Trombetti-Zhou codes are not linear over the entire field $\mathbb{F}_{q^{2n}}$, but only over its subfield $\mathbb{F}_{q^{n}}$. Due to this lack of linearity, we introduce the notions of $\mathbb{F}_{q^{n}}$-generator matrix and $\mathbb{F}_{q^{n}}$-parity-check matrix for a generic $\mathbb{F}_{q^{n}}$-linear rank-metric code over $\mathbb{F}_{q^{rn}}$ in analogy with the roles that generator and parity-check matrices play in the context of linear codes. Accordingly, we present an $\mathbb{F}_{q^n}$-generator matrix and $\mathbb{F}_{q^n}$-parity-check matrix for Trombetti-Zhou codes as evaluation codes over an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^{2n}}$. This relies on the choice of a particular basis called \emph{trace almost dual basis}. Subsequently, denoting by $d$ the minimum distance of the code, we show that if the rank weight $t$ of the error vector is strictly smaller than $\frac{d-1}{2}$, the syndrome-based decoding of Trombetti-Zhou codes can be converted to the decoding of Gabidulin codes of dimension one larger. On the other hand, when $t=\frac{d-1}{2}$, we reduce the decoding to determining the rank of a certain matrix. The complexity of the proposed decoding for Trombetti-Zhou codes is also discussed.

Decoding Trombetti-Zhou codes: a new syndrome-based decoding approach

TL;DR

The paper develops a syndrome-based decoder for Trombetti–Zhou (TZ) rank-metric codes, which are -linear over and not fully -linear. It introduces -generator and parity-check matrices built from a trace almost dual basis, enabling a syndrome-based decoding framework that, for error rank , reduces to decoding Gabidulin codes of dimension larger, while for the critical case it reduces to a structured rank-determination problem. The method constructs an error-span polynomial , recovers an error decomposition , and reconstructs the original codeword; a detailed complexity analysis shows a cubic complexity in the half-length over . The work thus provides a natural decoder for TZ codes that can exploit erasures/deviations and advances the understanding of subfield-linear rank-metric codes with practical decoding guarantees. Open problems include extending beyond the strict unique-decoding-radius constraint and improving efficiency relative to interpolation-based decoders.

Abstract

In 2019, Trombetti and Zhou introduced a new family of -linear Maximum Rank Distance (MRD) codes over . For such codes we propose a new syndrome-based decoding algorithm. It is well known that a syndrome-based decoding approach relies heavily on a parity-check matrix of a linear code. Nonetheless, Trombetti-Zhou codes are not linear over the entire field , but only over its subfield . Due to this lack of linearity, we introduce the notions of -generator matrix and -parity-check matrix for a generic -linear rank-metric code over in analogy with the roles that generator and parity-check matrices play in the context of linear codes. Accordingly, we present an -generator matrix and -parity-check matrix for Trombetti-Zhou codes as evaluation codes over an -basis of . This relies on the choice of a particular basis called \emph{trace almost dual basis}. Subsequently, denoting by the minimum distance of the code, we show that if the rank weight of the error vector is strictly smaller than , the syndrome-based decoding of Trombetti-Zhou codes can be converted to the decoding of Gabidulin codes of dimension one larger. On the other hand, when , we reduce the decoding to determining the rank of a certain matrix. The complexity of the proposed decoding for Trombetti-Zhou codes is also discussed.

Paper Structure

This paper contains 13 sections, 24 theorems, 183 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Outline of the paper
  4. Preliminaries
  5. Overview and new insights into Trombetti-Zhou codes
  6. Decoding Trombetti-Zhou codes: a new syndrome-based decoding approach
  7. Syndrome Calculation
  8. Determine The Error Span Polynomial
  9. Finding the Root Space of $\boldsymbol{\Lambda(x)}$
  10. Determining the Error
  11. Determine The Error Span Polynomial for $\mathbf{t=n-\frac{k}{2}}$.
  12. Complexity Analysis
  13. Conclusions and open problems

Key Result

Theorem 2.7

Let $\mathcal{C}\subseteq\mathbb{F}_{q^m}^\ell$ be a rank-metric code with $\lvert\mathcal{C}\rvert\geq 2$ and minimum distance $d$. Then If the parameters of $\mathcal{C}$ meet the bound, then the code is called a Maximum Rank Distance (MRD) code.

Figures (1)

  • Figure 1: Relationships among the rank-metric codes $\mathcal{D}_k(\gamma)$,$\mathcal{D}_k(\gamma)^{\perp}$, $\mathcal{TZ}_k(\gamma)[\underline{\lambda}]$ and $\mathcal{TZ}_k(\gamma)[\underline{\lambda}]^{\mathrel{\hbox{o}rigin=c]{90}{\hbox{$\models$}}}}$.

Theorems & Definitions (62)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Example 2.9
  • Theorem 2.10
  • ...and 52 more