Computer Science > Information Theory
[Submitted on 7 Sep 2021 (v1), last revised 27 May 2022 (this version, v13)]
Title:List-decodable Codes and Covering Codes
View PDFAbstract:The list-decodable code has been an active topic in theoretical computer this http URL are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. 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. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only on the radius. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the sizes 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. A generalized Singleton upper bound for average-radius list-decodable codes is also given from our general covering code upper bound. 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 list-decodable deletion codes,list-decodable sum-rank-metric codes and list decodable permutation codes. Some new better results about non-list-decodability of rank-metric codes, subspace codes, sum-rank-metric codes and permutation codes with various metrics are obtained.
Submission history
From: Hao Chen [view email][v1] Tue, 7 Sep 2021 02:04:41 UTC (13 KB)
[v2] Fri, 17 Sep 2021 13:30:00 UTC (22 KB)
[v3] Sun, 3 Oct 2021 03:20:08 UTC (23 KB)
[v4] Tue, 12 Oct 2021 04:03:10 UTC (24 KB)
[v5] Tue, 19 Oct 2021 07:42:22 UTC (25 KB)
[v6] Mon, 25 Oct 2021 16:05:38 UTC (25 KB)
[v7] Fri, 29 Oct 2021 00:42:26 UTC (25 KB)
[v8] Fri, 19 Nov 2021 10:16:02 UTC (25 KB)
[v9] Mon, 22 Nov 2021 22:15:57 UTC (26 KB)
[v10] Thu, 25 Nov 2021 23:02:44 UTC (28 KB)
[v11] Wed, 5 Jan 2022 11:15:03 UTC (30 KB)
[v12] Mon, 17 Jan 2022 23:32:54 UTC (31 KB)
[v13] Fri, 27 May 2022 22:40:14 UTC (31 KB)
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