Explicit List-Decodable Linearized Reed-Solomon Subspace Codes via Subspace Designs
Kuo Shang, Chen Yuan, Ruiqi Zhu
TL;DR
This work addresses explicit construction of sum-rank metric codes that can be efficiently list-decoded beyond the unique-decoding radius. It introduces an interpolation-based linear-algebraic framework for LRS codes and extends it to folded LRS codes, achieving explicit $\,\mathbb{F}_h$-linear subcodes with rates near capacity $R\approx 1$ and decoding radii approaching the list-decoding limit, with provably small list sizes. The key innovation is combining a structured, low-dimensional affine solution space (periodic subspaces) with subspace designs or subspace-evasive sets to bound the intersection and hence the list size, yielding explicit positive-rate sum-rank codes that admit efficient list decoding. This provides a general, explicit construction paradigm for efficiently list-decodable codes under the sum-rank metric, with potential impact on network coding, distributed storage, and quantum-resistant cryptography.
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 $ρ$ of errors in the sum-rank metric with rate $1-ρ-\varepsilon$, for any desired $ρ\in (0,1)$ 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.