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.Our results give exponential improvements on the recent generalized Singleton upper bound in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. Even for the list size $L=1$ case our covering code upper bounds give highly non-trivial upper bounds on the sizes of codes with the given minimum distance.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.苏丹和V.Guruswami的开创性论文发表以来,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被考虑。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中,列表标记标记代码的上边界限是新的简单但强有力的。覆盖代码上限的通用框可以适用于在中心依赖球量、子空间测量度测量、覆盖封面度测量度度测量、覆盖半径、对覆盖范围的任何好的上层界限代码都与列表标记代码的大小有着密切的联系。当代码长度大时,当清单的列表可删除度、可删除式列表的代码的上端框时,即使清单的大小为 $L=1美元,我们覆盖代码的上层列表中的部分约束值, 也能够将列表的上层的内序代码作为高级码。