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{align}

\operatorname {Pr} [ Y = 0 | X = 0 ] &= 1 \\ \operatorname {Pr} [ Y = 0 | X = 1 ] &= p \\ \operatorname {Pr} [ Y = 1 | X = 0 ] &= 0 \\ \operatorname {Pr} [ Y = 1 | X = 1 ] &= 1 - p \end{align}$$

Capacity
The channel capacity $$\mathsf{cap}(\mathbb{Z})$$ of the Z-channel $$\mathbb{Z}$$ 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:


 * $$\mathsf{cap}(\mathbb{Z}) = \mathsf{H}\left(\frac{1}{1+2^{\mathsf{s}(p)}}\right) - \frac{\mathsf{s}(p)}{1+2^{\mathsf{s}(p)}} = \log_2(1{+}2^{-\mathsf{s}(p)}) = \log_2\left(1+(1-p) p^{p/(1-p)}\right) $$

where $$\mathsf{s}(p) = \frac{\mathsf{H}(p)}{1-p}$$ for the binary entropy function $$\mathsf{H}(\cdot)$$.

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:


 * $$\alpha = 1 - \frac{1}{(1-p)(1+2^{\mathsf{H}(p)/(1-p)})},$$

For small p, the capacity is approximated by


 * $$ \mathsf{cap}(\mathbb{Z}) \approx 1- 0.5 \mathsf{H}(p) $$

as compared to the capacity $$1{-}\mathsf{H}(p)$$ of the binary symmetric channel with crossover probability p.


 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Calculation \max_\alpha\{\mathsf{H}(Y) - \mathsf{H}(Y \mid X)\} = \max_\alpha\Bigl\{\mathsf{H}(Y) - \sum_{x \in \{0,1\}}\mathsf{H}(Y \mid X = x) \mathsf{Prob}\{X = x\}\Bigr\} $$
 * $$\mathsf{cap}(\mathbb{Z}) =
 * $$\mathsf{cap}(\mathbb{Z}) =
 * $$=\max_\alpha\{\mathsf{H}((1-\alpha)(1-p)) - \mathsf{H}(Y \mid X = 1) \mathsf{Prob}\{X = 1\} \}$$
 * $$=\max_\alpha\{\mathsf{H}((1-\alpha)(1-p)) - (1-\alpha)\mathsf{H}(p) \},$$

To find the maximum we differentiate
 * $$\frac{d}{d\alpha}\mathsf{cap}(\mathbb{Z}) = -(1-p)\log_2\left(\frac{1-(1-\alpha)(1-p)}{(1-\alpha)(1-p)}\right)+\mathsf{H}(p)$$

And we see the maximum is attained for
 * $$\alpha = 1 - \frac{1}{(1-p)(1+2^{\mathsf{H}(p)/(1-p)})},$$

yielding the following value of $$\mathsf{cap}(\mathbb{Z})$$ as a function of p
 * $$\mathsf{cap}(\mathbb{Z}) = \mathsf{H}\left(\frac{1}{1+2^{\mathsf{s}(p)}}\right) - \frac{\mathsf{s}(p)}{1+2^{\mathsf{s}(p)}} = \log_2(1{+}2^{-\mathsf{s}(p)}) = \log_2\left(1+(1-p) p^{p/(1-p)}\right) \; \textrm{ where } \; \mathsf{s}(p) = \frac{\mathsf{H}(p)}{1-p}.$$


 * }

For any p, $$\alpha<0.5$$ (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 $$\frac{1}{e}$$.

Bounds on the size of an asymmetric-error-correcting code
Define the following distance function $$\mathsf{d}_A(\mathbf{x}, \mathbf{y})$$ on the words $$\mathbf{x}, \mathbf{y} \in \{0,1\}^n$$ of length n transmitted via a Z-channel
 * $$\mathsf{d}_A(\mathbf{x}, \mathbf{y}) \stackrel{\vartriangle}{=} \max\left\{ \big|\{i \mid x_i = 0, y_i = 1\}\big|, \big|\{i \mid x_i = 1, y_i = 0\}\big| \right\}.$$

Define the sphere $$V_t(\mathbf{x})$$ of radius t around a word $$\mathbf{x} \in \{0,1\}^n$$ of length n as the set of all the words at distance t or less from $$\mathbf{x}$$, in other words,
 * $$V_t(\mathbf{x}) = \{\mathbf{y} \in \{0, 1\}^n \mid \mathsf{d}_A(\mathbf{x}, \mathbf{y}) \leq t\}.$$

A code $$\mathcal{C}$$ of length n is said to be t-asymmetric-error-correcting if for any two codewords $$\mathbf{c}\ne \mathbf{c}' \in \{0,1\}^n$$, one has $$V_t(\mathbf{c}) \cap V_t(\mathbf{c}') = \emptyset$$. 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,
 * $$M(n,t) \leq \frac{2^{n+1}}{\sum_{j = 0}^t{\left( \binom{\lfloor n/2\rfloor}{j}+\binom{\lceil n/2\rceil}{j}\right)}}.$$

The constant-weight code bound. For n > 2t ≥ 2, let the sequence B0, B1, ..., Bn-2t-1 be defined as
 * $$B_0 = 2, \quad B_i = \min_{0 \leq j < i}\{ B_j + A(n{+}t{+}i{-}j{-}1, 2t{+}2, t{+}i)\}$$ for $$i > 0$$.

Then $$M(n,t) \leq B_{n-2t-1}.$$