Codes for the Z-Channel

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.