Codes for the Z-channel
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This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction τ takes as input a length-n binary codeword and injects in an adversarial manner nτ 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 τ has exponential (in n) size if τ is less than a critical value that we call the Plotkin point and has constant size if τ 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 ϵ-above the Plotkin point has size Θ_L(ϵ^-3/2) for any L-1≥1. We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.
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