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。和秩度量在网络编码、局部可修复码、空时编码以及抗量子密码学中均有应用。线性化Reed-Solomon（LRS）码是和秩度量下的Reed-Solomon码类比，并严格推广了Reed-Solomon码与Gabidulin码。本文中，我们在任意域$\mathbb{F}_h$上构造了一个显式的$\mathbb{F}_h$-线性和秩度量码族。对于任意给定的$\rho\in (0,1)$和$\varepsilon>0$，我们的构造能够实现高效列表解码，在和秩度量下可纠正高达$\rho$比例的差错，且码率为$1-\rho-\varepsilon$。这些码是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$\mathbb{F}_h$-子空间上获得，且解码列表大小以$h^{\mathrm{poly}(1/\varepsilon)}$为界。在标准LRS框架之外，我们进一步将线性代数解码框架推广到折叠线性化Reed-Solomon（FLRS）码。我们证明了折叠求值满足适当的插值条件，且对应的解空间构成一个低维、结构化的仿射子空间。这一结构使得列表大小能够得到有效控制，并首次给出了显式的、具有正码率的FLRS子码，可在唯一解码半径之外实现高效列表解码。据我们所知，这也是首个显式构造的正码率和秩度量码族，能够在唯一解码半径之外实现高效列表解码，从而为在和秩度量下构造高效可解码码提供了一个新的通用框架。