List-decodable Codes and Covering Codes
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by
Hao Chen
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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