The Annals of Mathematical Statistics

Exponential Bounds on the Probability of Error for a Discrete Memoryless Channel

Samuel Kotz

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Abstract

In a paper by Blackwell, Breiman and Thomasian [1, Theorem 3] the following theorem is proved: For any integer $n$ and for any $0 < \epsilon \leqq \frac{1}{2}$, such that $C - \epsilon \geqq 0$ there exists a code for a discrete memoryless channel with length $N > e^{n(C - \epsilon)}$ and with a bound for the probability of error, $\bar\lambda = 2 \exp_e - \lbrack n\epsilon^2/(16ab)\rbrack$, where $C$ is the capacity of the channel and a and b are the numbers of elements in the input and output alphabets respectively. In this note we shall replace the bound $2 \exp_e\lbrack -n\epsilon^2/(16ab)\rbrack$ by the expression $2 \exp_e\{-n\epsilon^2/\lbrack g(c)(\log c)^\{2 - \delta\rbrack\}$, where $c = \min (a, b), g(c)$ is a positive monotonically decreasing function of $c, g(c) < 16$ for all $c \geqq 3$ and approaches 2 asymptotically as $c \rightarrow \infty$, and $\delta > 0$ depends on $\epsilon$ and $c$ and tends to 0 as either $c \rightarrow \infty$ or $\epsilon \rightarrow 0$.

Article information

Source
Ann. Math. Statist., Volume 32, Number 2 (1961), 577-582.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177705062

Digital Object Identifier
doi:10.1214/aoms/1177705062

Mathematical Reviews number (MathSciNet)
MR129059

Zentralblatt MATH identifier
0096.33703

JSTOR
links.jstor.org

Citation

Kotz, Samuel. Exponential Bounds on the Probability of Error for a Discrete Memoryless Channel. Ann. Math. Statist. 32 (1961), no. 2, 577--582. doi:10.1214/aoms/1177705062. https://projecteuclid.org/euclid.aoms/1177705062


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