Abstract:
This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction \tau takes as input a length- n bina...Show MoreMetadata
Abstract:
This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction \tau takes as input a length- n binary codeword and injects in an adversarial manner up to n\tau asymmetric errors, i.e., errors that only zero out bits but do not flip 0’s to 1’s. It is known that the largest (L-1) -list-decodable code for the Z-channel with error fraction \tau has exponential size (in n ) if \tau is less than a critical value that we call the (L-1) -list-decoding Plotkin point and has constant size if \tau is larger than the threshold. The (L-1) -list-decoding Plotkin point is known to be L^{-({1}/{L-1})} - L^{-({L}/{L-1})} , which equals 1/4 for unique-decoding with L-1=1 . In this paper, we derive various results for the size of the largest codes above and below the list-decoding Plotkin point. In particular, we show that the largest (L-1) -list-decodable code \varepsilon -above the Plotkin point, for any given sufficiently small positive constant \varepsilon >0 , has size \Theta _{L}(\varepsilon ^{-3/2}) for any L-1\ge 1 . We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.
Published in: IEEE Transactions on Information Theory ( Volume: 69, Issue: 10, October 2023)