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.

翻译：和秩度量是汉明度量和秩度量的混合体。和秩度量在网络编码、局部可修复码、空时编码及抗量子密码学中具有重要应用。线性化Reed-Solomon（LRS）码是和秩度量下的Reed-Solomon码类比，严格推广了Reed-Solomon码与Gabidulin码。本文针对任意域$\mathbb{F}_h$构造了一个显式的$\mathbb{F}_h$-线性和秩度量码族。该构造能以速率$1-ρ-\varepsilon$实现和秩度量下最高$ρ$比例错误的高效列表解码，其中$ρ\in (0,1)$与$\varepsilon>0$可任意选取。所构造的码是LRS码的子码，通过将消息多项式限制在由子空间设计导出的$\mathbb{F}_h$-子空间中获得，解码列表大小以$h^{\mathrm{poly}(1/\varepsilon)}$为界。在标准LRS框架之外，我们进一步将线性代数解码框架扩展至折叠线性化Reed-Solomon（FLRS）码。我们证明折叠求值满足适当的插值条件，且对应的解空间构成低维结构化仿射子空间。这种结构能有效控制列表大小，并首次得到可高效列表解码超越唯一解码半径的显式正速率FLRS子码。据我们所知，这也是首个超越唯一解码半径且支持高效列表解码的正速率和秩度量码的显式构造，从而为构建和秩度量下高效可解码码提供了新的通用框架。