Explicit List-Decodable Linearized Reed-Solomon and Folded Linearized Reed-Solomon Subcodes

2026-02-05

AI Summary

This work addresses the absence of explicit constant-rate code constructions in the rank metric that admit efficient list decoding beyond the unique decoding radius. By imposing subspace design constraints on message polynomials, the authors construct a linearized Reed–Solomon subcode and extend it to a folded variant, yielding the first explicit code family satisfying these requirements. Leveraging linear-algebraic decoding techniques combined with structural analysis of affine subspaces, the proposed scheme achieves efficient list decoding for any error fraction ρ, attaining a rate approaching 1−ρ with list size h^{poly(1/ε)}. This establishes a general and efficient list decoding framework tailored to the rank metric.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Non-convex Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The sum-rank metric is the mixture of the Hamming and rank metrics. The sum-rank metric found its application in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. Linearized Reed-Solomon (LRS) codes are the sum-rank analogue of Reed-Solomon codes and strictly generalize both Reed-Solomon and Gabidulin codes. In this work, we construct an explicit family of $\mathbb{F}_h$-linear sum-rank metric codes over arbitrary fields $\mathbb{F}_h$. Our construction enables efficient list decoding up to a fraction $\rho$ of errors in the sum-rank metric with rate $1-\rho-\varepsilon$, for any desired $\rho \in (0,1)$ and $\varepsilon>0$. Our codes are subcodes of LRS codes, obtained by restricting message polynomials to an $\mathbb{F}_h$-subspace derived from subspace designs, and the decoding list size is bounded by $h^{\mathrm{poly}(1/\varepsilon)}$. Beyond the standard LRS setting, we further extend our linear-algebraic decoding framework to folded Linearized Reed-Solomon (FLRS) codes. We show that folded evaluations satisfy appropriate interpolation conditions and that the corresponding solution space forms a low-dimensional, structured affine subspace. This structure enables effective control of the list size and yields the first explicit positive-rate FLRS subcodes that are efficiently list decodable beyond the unique-decoding radius. To the best of our knowledge, this also constitutes the first explicit construction of positive-rate sum-rank metric codes that admit efficient list decoding beyond the unique decoding radius, thereby providing a new general framework for constructing efficiently decodable codes under the sum-rank metric.

Problem

• sum-rank metric
• list decoding
• Linearized Reed-Solomon codes
• explicit construction
• beyond unique decoding radius

Innovation

• list decoding
• sum-rank metric
• Linearized Reed-Solomon codes
• subspace designs
• folded codes