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

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.