A lower bound on the list-decodability of insdel codes

• Main Authors: Liu, Shu, Tjuawinata, Ivan, Xing, Chaoping
• Other Authors: School of Computer Science and Engineering
• Format: Article
• Language: English
• Published: 2023
• Subjects: Science::Mathematics::Applied mathematics::Information theory; Engineering::Computer science and engineering::Data::Coding and information theory; Codes; Information Theory; List Decoding; Insertion and Deletion Errors
• Online Access: https://hdl.handle.net/10356/169889
• Institution: Nanyang Technological University

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, list size and the alphabet size. 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 is not list-decodable with a given list size, then the list decoding radius is lower bounded by a bound involving the list size and Levenshtein distance. In other words, if the list decoding radius is less than this lower bound, the code must be list-decodable with the given list size. 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.