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

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 $\epsilon \rightarrow 0$.