Z-channel (information theory)

In coding theory and information theory, a Z-channel or binary asymmetric channel is a communications channel used to model the behaviour of some data storage systems.

A Z-channel is a channel with binary input and binary output, where each 0 bit is transmitted correctly, but each 1 bit has probability p of being transmitted incorrectly as a 0, and probability 1–p of being transmitted correctly as a 1.

In other words, if X and Y are the random variables describing the probability distributions of the input and the output of the channel, respectively, then the crossovers of the channel are characterized by the conditional probabilities:[1] The channel capacity

{\displaystyle {\mathsf {cap}}(\mathbb {Z} )}

with the crossover 1 → 0 probability p, when the input random variable X is distributed according to the Bernoulli distribution with probability

for the occurrence of 0, is given by the following equation: where

for the binary entropy function

This capacity is obtained when the input variable X has Bernoulli distribution with probability

1 − α

of value 1, where: For small p, the capacity is approximated by as compared to the capacity

of the binary symmetric channel with crossover probability p. To find the maximum we differentiate And we see the maximum is attained for yielding the following value of

α > 0.5

(i.e. more 0s should be transmitted than 1s) because transmitting a 1 introduces noise.

[2] Define the following distance function

of length n transmitted via a Z-channel Define the sphere

of radius t around a word

of length n as the set of all the words at distance t or less from

, in other words, A code

of length n is said to be t-asymmetric-error-correcting if for any two codewords

the maximum number of codewords in a t-asymmetric-error-correcting code of length n. The Varshamov bound.

For n≥1 and t≥1, The constant-weight[clarification needed] code bound.

For n > 2t ≥ 2, let the sequence B0, B1, ..., Bn-2t-1 be defined as Then