List-decodable Codes and Covering Codes

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, 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. Even for the list size $L=1$ case our covering code upper bounds give highly non-trivial upper bounds on the sizes of codes with the given minimum distance. 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 and list-decodable sum-rank-metric codes. Some new better results about non-list-decodability of rank-metric codes, subspace codes and sum-rank-metric codes are obtained.