For codes equipped with metrics such as Hamming metric, symbol pair metric or cover metric, the Johnson bound guarantees list-decodability of such codes. That is, the Johnson bound provides a lower bound on the list-decoding radius of a code in terms of its relative minimum distance $\delta$, list size $L$ and the alphabet size $q.$ For study of list-decodability of codes with insertion and deletion errors (we call such codes insdel codes), it is natural to ask the open problem whether there is also a Johnson-type bound. The problem was first investigated by Wachter-Zeh and the result was amended by Hayashi and Yasunaga where a lower bound on the list-decodability for insdel codes was derived. The main purpose of this paper is to move a step further towards solving the above open problem. In this work, we provide a new lower bound for the list-decodability of an insdel code. As a consequence, we show that unlike the Johnson bound for codes under other metrics that is tight, the bound on list-decodability of insdel codes given by Hayashi and Yasunaga is not tight. Our main idea is to show that if an insdel code with a given Levenshtein distance $d$ is not list-decodable with list size $L$, then the list decoding radius is lower bounded by a bound involving $L$ and $d$. In other words, if the list decoding radius is less than this lower bound, the code must be list-decodable with list size $L$. At the end of the paper we use such bound to provide an insdel-list-decodability bound for various well-known codes, which has not been extensively studied before.

翻译：对于配备汉明度量、符号对度量或覆盖度量的码，约翰逊界保证了此类码的列表可译性。即，约翰逊界根据码的相对最小距离$\delta$、列表大小$L$和字母表大小$q$，给出了码的列表译码半径的下界。为了研究具有插入和删除错误的码（我们称此类码为插入删除码）的列表可译性，自然会提出一个开放问题：是否存在类似的约翰逊型界？该问题最初由Wachter-Zeh研究，其后Hayashi和Yasunaga修正了结果，推导出插入删除码列表可译性的一个下界。本文的主要目的是在解决上述开放问题的道路上迈出进一步。在这项工作中，我们为插入删除码的列表可译性提供了一个新的下界。作为推论，我们表明与在其他度量下紧致的约翰逊界不同，Hayashi和Yasunaga给出的插入删除码列表可译性界并非紧致。我们的主要思想是证明：如果一个具有给定莱温斯坦距离$d$的插入删除码在列表大小为$L$时不可列表译码，那么列表译码半径由一个涉及$L$和$d$的界给出下界。换句话说，如果列表译码半径小于此下界，则该码必定在列表大小为$L$时是可列表译码的。在论文末尾，我们利用该界为多种已知码提供了插入删除列表可译性界，这一方向此前尚未得到广泛研究。