In this paper, we establish the list-decoding capacity theorem for sum-rank metric codes. This theorem implies the list-decodability theorem for random general sum-rank metric codes: Any random general sum-rank metric code with a rate not exceeding the list-decoding capacity is $\left(\rho,O\left(1/\epsilon\right)\right)$-list-decodable with high probability, where $\rho\in\left(0,1\right)$ represents the error fraction and $\epsilon>0$ is referred to as the capacity gap. For random $\mathbb{F}_q$-linear sum-rank metric codes by using the same proof approach we demonstrate that any random $\mathbb{F}_q$-linear sum-rank metric code with a rate not exceeding the list-decoding capacity is $\left(\rho,\exp\left(O\left(1/\epsilon\right)\right)\right)$-list-decodable with high probability, where the list size is exponential at this stage due to the high correlation among codewords in linear codes. To achieve an exponential improvement on the list size, we prove a limited correlation property between sum-rank metric balls and $\mathbb{F}_q$-subspaces. Ultimately, we establish the list-decodability theorem for random $\mathbb{F}_q$-linear sum-rank metric codes: Any random $\mathbb{F}_q$-linear sum-rank metric code with rate not exceeding the list decoding capacity is $\left(\rho, O\left(1/\epsilon\right)\right)$-list-decodable with high probability. For the proof of the list-decodability theorem of random $\mathbb{F}_q$-linear sum-rank metric codes our proof idea is inspired by and aligns with that provided in the works \cite{Gur2010,Din2014,Gur2017} where the authors proved the list-decodability theorems for random $\mathbb{F}_q$-linear Hamming metric codes and random $\mathbb{F}_q$-linear rank metric codes, respectively.

翻译：本文建立了和-秩度量码的列表译码容量定理。该定理蕴含了随机一般和-秩度量码的列表可译性定理：任何码率不超过列表译码容量的随机一般和-秩度量码，以高概率是 $\left(\rho,O\left(1/\epsilon\right)\right)$-列表可译的，其中 $\rho\in\left(0,1\right)$ 表示错误比例，$\epsilon>0$ 被称为容量间隙。对于随机 $\mathbb{F}_q$-线性和-秩度量码，采用相同的证明方法，我们证明了任何码率不超过列表译码容量的随机 $\mathbb{F}_q$-线性和-秩度量码，以高概率是 $\left(\rho,\exp\left(O\left(1/\epsilon\right)\right)\right)$-列表可译的，其中由于线性码中码字间的高度相关性，此时的列表大小是指数级的。为了实现对列表大小的指数级改进，我们证明了和-秩度量球与 $\mathbb{F}_q$-子空间之间的有限相关性性质。最终，我们建立了随机 $\mathbb{F}_q$-线性和-秩度量码的列表可译性定理：任何码率不超过列表译码容量的随机 $\mathbb{F}_q$-线性和-秩度量码，以高概率是 $\left(\rho, O\left(1/\epsilon\right)\right)$-列表可译的。在证明随机 $\mathbb{F}_q$-线性和-秩度量码的列表可译性定理时，我们的证明思路受到并遵循了文献 \cite{Gur2010,Din2014,Gur2017} 中提供的思路，这些文献的作者分别证明了随机 $\mathbb{F}_q$-线性汉明度量码和随机 $\mathbb{F}_q$-线性秩度量码的列表可译性定理。