Last updated 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.
A Z-channel or binary asymmetric channel is a communications channel model used in coding theory and information theory to represent certain data storage systems. In a Z-channel, 0 bits are always transmitted correctly, but 1 bits may be corrupted and received as 0s with some probability.
This asymmetric error pattern, where errors only occur in one direction (1→0 but never 0→1), makes Z-channels particularly relevant for modeling storage technologies with intrinsic recording asymmetries. For example, in some flash memory systems, stored 1s may degrade over time and be read as 0s, but stored 0s remain stable.
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:[1]
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:
yielding the following value of as a function of p
For any , (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
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
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 (seconded.). 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.
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