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

2025-11-28

AI Summary

Trombetti–Zhou (TZ) codes are only $mathbb{F}_{q^n}$-linear—not $mathbb{F}_{q^{2n}}$-linear—rendering conventional parity-check matrix decoding inapplicable. Method: We propose the first syndrome decoding algorithm for TZ codes, leveraging an $mathbb{F}_{q^n}$-parity-check matrix and a trace almost-dual basis. By revealing their $mathbb{F}_{q^n}$-linear evaluation code structure, we reduce decoding to either Gabidulin decoding (for rank errors $t < (d-1)/2$) or matrix rank determination (for $t = (d-1)/2$). Contribution/Results: This approach circumvents the traditional requirement of full-field linearity, establishing the first systematic decoding framework for $mathbb{F}_{q^n}$-linear rank-metric codes. We provide a complete complexity analysis and significantly extend the theoretical decodability boundary for nonlinear rank-metric codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Developing syndrome-based decoding for nonlinear Trombetti-Zhou MRD codes
• Constructing parity-check matrices for rank-metric codes over subfields
• Converting decoding to Gabidulin codes when error weight is small

Innovation

• Syndrome-based decoding for non-linear Trombetti-Zhou codes
• Using subfield generator and parity-check matrices
• Converting decoding to Gabidulin codes for efficiency