列表可变编码和覆盖编码 (List-decodable Codes and Covering Codes)

The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. List-decodable codes are also considered in rank-metric, subspace metric, cover-metric, pair metric and insdel metric settings. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes of finite metric spaces. The general covering code upper bounds can apply to the case when the volumes of the balls depend on the centers, not only on the radius case. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the size of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes on general finite metric spaces. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are very large. The generalized Singleton upper bound for average-radius list-decodable codes is given. The asymptotic forms of covering code bounds can partially recover the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman in Hamming metric setting. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes. We apply our general covering code upper bounds for list-decodable rank-metric codes, list-decodable subspace codes, list-decodable insertion codes and list-decodable deletion codes. Some new better results about non-list-decodability of rank-metric codes and subspace codes are obtained.

翻译：自1997-1998年M. Sudan和V. Guruswami的开创性论文以来,列表标记代码一直是理论计算机科学的一个活跃话题。列表标记代码也可以在等级测量、子空间测量、覆盖度测量、对称度测量和内嵌度设置中加以考虑。在本文中,我们显示比率、列表标记半径和列表大小与覆盖代码的经典主题密切相关。我们证明,在基于各种限定度空间代码的通用限值标准空间中,列表标记可辨识代码具有新的简单但强大的上界。当球的量依赖中心时,覆盖代码上限的通用代码可以适用于案件。然后,在覆盖半径的半径度、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对内、对内、对内、对面、对内、对面、对内、对面、对内、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对内、对内、对面、对面、对内、对内、对内、对内、对内、对内、对内、对内、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对面、对内、对内、对面、对内、对内、对内、对内、对内、对内、对内、对面、对内、对内、对内、对内、对内、