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. There are general result about the Johnson radius and the list-decoding capacity theorem for random codes. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. 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 simple but strong upper bounds for list-decodable codes based on various covering codes. Then any good upper bound on the covering radius 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. Our covering code upper bounds for $(d,1)$ list decodable codes give highly non-trivial upper bounds on the sizes of codes with the given minimum Hamming distances. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020, when the code lengths are very large. The asymptotic forms of covering code bounds can partially recover the list-decoding capacity theorem, the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes.
翻译:自1997-1998年M.苏丹和V.Guruswami的创刊论文发表以来,列表标记代码一直是理论计算机科学的一个活跃话题。关于约翰逊半径和随机代码的列表解码能力理论,总的结果是约翰逊半径和列表解码能力理论。但是,关于利率、列表标记半径和列表标记代码列表大小的一般限制结果很少。在本文中,我们显示,比率、列表可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,基于各种覆盖代码的列表可标记代码有新的简单但强的上界。然后,任何覆盖半径的正确上界都意味着列表可辨码的大小有一个良好的上界。因此,代码的列表分解性是覆盖代码视图的强烈限制。我们覆盖的$(d),$(d)的可标记半径和列表的编码使代码大小的高度非三维的上限与设定的最低明距离。我们的结果使最近普及的单项单项编码列表的指数改进了指数的指数性, 也表明, 将STOO-co-co-co-co-co-co-co-co-co-co-co-co-co-co-la-la-la-la-la-lax-suder-suder-la-la-la-la-suder-la-la-la-la-la-la-la-la-la-de-la-la-la-suol-suol-suder-de-de-de-de-de-de-de-de-de-de-al-de-de-de-al-de-de-de-de-de-de-de-de-de-de-de-de-de-de-de-de-de-de-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-de-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al