List-decodable Codes and Covering Codes
Hao Chen
TL;DR
This paper explores the relationship between list-decodable codes and covering codes, providing new bounds and improvements for various metrics and code types, advancing theoretical understanding in coding theory.
Contribution
It establishes strong upper bounds for list-decodable codes using covering code theory, improving existing bounds and extending results to multiple code metrics and types.
Findings
Exponential improvements on the generalized Singleton bound for large code lengths.
New bounds for list-decodable rank-metric, subspace, and permutation codes.
Insights into non-list-decodability of certain codes across various metrics.
Abstract
The list-decodable code has been an active topic in theoretical computer science.There 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,…
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Cooperative Communication and Network Coding