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^{-\frac{1}{L-1}} - L^{-\frac{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 $\epsilon$-above the Plotkin point, {for any given sufficiently small positive constant $ \epsilon>0 $,} has size $\Theta_L(\epsilon^{-3/2})$ for any $L-1\ge1$. We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.

翻译：本文系统研究了Z信道编码的组合特性。考虑误差比例为τ的Z信道，该信道以长度为n的二进制码字作为输入，并以对抗方式注入至多nτ个非对称错误——即仅将1翻转为0而不会将0翻转为1的错误。已知当τ小于某个临界值（定义为(L-1)-列表解码普洛特金点）时，Z信道最大(L-1)-列表可解码码具有指数规模（关于n），当τ大于该阈值时则具有常数规模。当前已知(L-1)-列表解码普洛特金点为L^{-1/(L-1)} - L^{-L/(L-1)}，当L-1=1时退化为唯一解码对应的1/4。本文针对列表解码普洛特金点上下区域最大码的规模推导了若干结论。特别地，我们证明：对于任意充分小的正常数ε>0，在普洛特金点上方ε范围内的最大(L-1)-列表可解码码的规模为Θ_L(ε^{-3/2})（对任意L-1≥1成立）。我们还建立了普洛特金点下方码指数规模的上界与下界。