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.

The Z-channel sees each 0 bit of a message transmitted correctly always and each 1 bit transmitted correctly with probability 1–p, due to noise across the transmission medium.

Definition

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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]

 

Capacity

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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 0 and   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.

For any p,   (i.e. more 0s should be transmitted than 1s) because transmitting a 1 introduces noise. As  , the limiting value of   is  .[2]

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

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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[clarification needed] code bound. For n > 2t ≥ 2, let the sequence B0, B1, ..., Bn-2t-1 be defined as

  for  .

Then  

Notes

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  1. ^ MacKay (2003), p. 148.
  2. ^ a b MacKay (2003), p. 159.

References

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  • MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. ISBN 0-521-64298-1.
  • Kløve, T. (1981). "Error correcting codes for the asymmetric channel". Technical Report 18–09–07–81. Norway: Department of Informatics, University of Bergen.
  • Verdú, S. (1997). "Channel Capacity (73.5)". The electrical engineering handbook (second ed.). IEEE Press and CRC Press. pp. 1671–1678.
  • Tallini, L.G.; Al-Bassam, S.; Bose, B. (2002). On the capacity and codes for the Z-channel. Proceedings of the IEEE International Symposium on Information Theory. Lausanne, Switzerland. p. 422.