Decoding Algorithms for Tensor Codes

2026-04-17

AI Summary

This study addresses the problem of efficient decoding of generalized tensor codes under the tensor rank metric. Focusing on low-rank error patterns, it proposes a fiber-structured decoding strategy based on Gabidulin codes and extends the Loidreau–Overbeck method to higher-order tensors. By integrating fiber decomposition, slice space analysis, and decoding techniques for generalized maximum rank distance codes, the approach effectively corrects errors measured by tensor rank weight, achieving reliable error correction under various metrics bounded above by the tensor rank. This work represents the first systematic extension of advanced matrix-code decoding frameworks into the tensor domain, significantly advancing the theory of high-dimensional error-correcting 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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Tensor codes are a generalisation of matrix codes. Such codes are defined as subspaces of order-r tensors for which the ambient space is endowed with the tensor-rank as a metric. A class of these codes was introduced by Roth, who also outlined a decoding algorithm for low tensor-rank errors that can be generalised to an algorithm with exponential complexity in the decoding radius. They may be viewed as a generalisation of the well-known Delsarte-Gabidulin-Roth maximum rank distance codes. We study a generalised class of these codes. We investigate their properties and outline decoding techniques for different metrics that leverage their tensor structure. We first consider a fibre-wise decoding approach, as each fibre of a codeword corresponds to a Gabidulin codeword. We then give a generalisation of Loidreau-Overbeck's decoding method that corrects errors with properties constrained by the dimensions of the slice spaces and fibre spaces. The metrics we consider are bounded from above by the tensor-rank metric, and therefore these algorithms also decode tensor-rank weight errors.

Problem

• tensor codes
• decoding algorithms
• tensor-rank metric
• error correction
• Gabidulin codes

Innovation

• tensor codes
• decoding algorithms
• fibre-wise decoding
• tensor-rank metric
• Gabidulin codes