Z-channel (information theory)


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

Definition

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:

Capacity

The channel capacity of the Z-channel 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 of having value 1 and of value 0, 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 as a function of p
For any p, because transmitting a 1 introduces noise. As, the limiting value of is.

Bounds on the size of an asymmetric-error-correcting code

Define the following distance function on the words 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, one has. Denote by 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 code bound.
For n > 2t ≥ 2, let the sequence B0, B1,..., Bn-2t-1 be defined as
Then