Singleton-type bounds for list-decoding and list-recovery, and related results1

List-decoding and list-recovery are important generalizations of unique decoding and receive considerable attention over the years. We study the optimal trade-off among the list-decoding (resp. list-recovery) radius, the list size, and the code rate, when the list size is constant and the alphabet size is large (both compared with the code length). We prove a new Singleton-type bound for list-decoding, which, for a wide range of parameters, is asymptotically tight up to a $1+o\left(1\right)$ factor. We also prove a Singleton-type bound for list-recovery, which is the first such bound in the literature. We apply these results to obtain near optimal lower bounds on the list size for list-decodable and list-recoverable codes with rates approaching capacity.

Moreover, we show that under some indivisibility condition of the parameters and over a sufficiently large alphabet, the largest list-decodable nonlinear codes can have much more codewords than the largest list-decodable linear codes. Such a large gap is not known to exist in unique decoding. We prove this by a novel connection between list-decoding and the notion of sparse hypergraphs in extremal combinatorics.

Lastly, we show that list-decodability or recoverability implies in some sense good unique decodability.