List-decodable Codes and Covering Codes release_pz6ebnba4fgafkw3galn7a4w64

by Hao Chen

Released as a article .

2021  

Abstract

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-Håstad-Sudan-Zuckerman. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes.
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Type  article
Stage   submitted
Date   2021-09-07
Version   v1
Language   en ?
arXiv  2109.02818v1
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