Z-channel (information theory)

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

Definition

A Z-channel (or a binary asymmetric channel) is a channel with binary input and binary output where the crossover 1 → 0 occurs with nonnegative probability p, whereas the crossover 0 → 1 never occurs. 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

Prob{Y = 0 | X = 0} = 1
Prob{Y = 0 | X = 1} = p
Prob{Y = 1 | X = 0} = 0
Prob{Y = 1 | X = 1} = 1p

Capacity

The 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 α for the occurrence of 0, is calculated as follows.

where is the binary entropy function.

The maximum is attained for

yielding the following value of as a function of p

For small p, the capacity is approximated by

as compared to the capacity of the binary symmetric channel with crossover probability p.

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

for .

Then

References

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