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.

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:

:\begin
\operatorname X = 0 &= 1 \\
\operatorname X = 1 &= p \\
\operatorname X = 0 &= 0 \\
\operatorname X = 1 &= 1 - p
\end

Capacity

The channel capacity $\mathsf(\mathbb)$ of the Z-channel $\mathbb$ with the crossover 1 → 0 probability ''p'', when the input random variable ''X'' is distributed according to the Bernoulli distribution with probability ''$\alpha$'' for the occurrence of 0, is given by the following equation:

:$\mathsf(\mathbb) = \mathsf\left(\frac\right) - \frac = \log_2(12^) = \log_2\left(1+(1-p) p^\right)$

where $\mathsf(p) = \frac$ for the binary entropy function $\mathsf(\cdot)$.

This capacity is obtained when the input variable ''X'' has Bernoulli distribution with probability $\alpha$ of having value 0 and $1-\alpha$ of value 1, where:

:$\alpha = 1 - \frac,$

For small ''p'', the capacity is approximated by

:$\mathsf(\mathbb) \approx 1- 0.5 \mathsf(p)$
as compared to the capacity $1\mathsf(p)$ of the binary symmetric channel with crossover probability ''p''.

:

For any $p > 0$, $\alpha>0.5$ (i.e. more 0s should be transmitted than 1s) because transmitting a 1 introduces noise. As $p\rightarrow 1$, the limiting value of $\alpha$ is $1-\frac$.

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

Define the following distance function $\mathsf_A(\mathbf, \mathbf)$ on the words $\mathbf, \mathbf \in \^n$ of length ''n'' transmitted via a Z-channel
:$\mathsf_A(\mathbf, \mathbf) \stackrel \max\left\.$
Define the sphere $V_t(\mathbf)$ of radius ''t'' around a word $\mathbf \in \^n$ of length ''n'' as the set of all the words at distance ''t'' or less from $\mathbf$, in other words,
:$V_t(\mathbf) = \.$
A code $\mathcal$ of length ''n'' is said to be ''t''-asymmetric-error-correcting if for any two codewords $\mathbf\ne \mathbf' \in \^n$, one has $V_t(\mathbf) \cap V_t(\mathbf') = \emptyset$. Denote by $M(n,t)$ the maximum number of codewords in a ''t''-asymmetric-error-correcting code of length ''n''.

The Varshamov bound.
For ''n''≥1 and ''t''≥1,
:$M(n,t) \leq \frac.$

The constant-weight code bound.
For ''n > 2t ≥ 2'', let the sequence ''B₀, B₁, ..., B_n-2t-1'' be defined as
:$B_0 = 2, \quad B_i = \min_\$ for $i > 0$.
Then $M(n,t) \leq B_.$

Notes

References

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* {{cite conference, last1=Tallini, first1=L.G. , last2=Al-Bassam , first2=S. , last3=Bose , first3=B. , title=On the capacity and codes for the Z-channel , conference=Proceedings of the IEEE International Symposium on Information Theory , location=Lausanne, Switzerland , year=2002 , page=422

Coding theory
Information theory
Inequalities (mathematics)