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} = 1−p

Capacity

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

{\mathsf {cap}}({\mathbb {Z}})=\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 \}}

=\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)\},

where ${\mathsf {H}}(\cdot )$ is the binary entropy function.

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 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.

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 ${\mathbf {c}}\neq {\mathbf {c}}'\in \{0,1\}^{n}$ , 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, ..., B_n-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 $M(n,t)\leq B_{{n-2t-1}}.$

References

T. Kløve, Error correcting codes for the asymmetric channel, Technical Report 18–09–07–81, Department of Informatics, University of Bergen, Norway, 1981.
S. Verdú, Channel Capacity, in The electrical engineering handbook, 2nd ed., IEEE Press and CRC Press, 1997, ch. 73.5, pp. 1671-1678.
L.G. Tallini, S. Al-Bassam, B. Bose, On the capacity and codes for the Z-channel, Proceedings of the IEEE International Symposium on Information Theory, Lausanne, Switzerland, 2002, p. 422.

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