A rate $R$ is called achievable for a discrete channel if there exists a sequence of $(n,M_{n})$ fixed-length codes $\mathcal{C}_{n}$ for the channel with $$\liminf_{n\to\infty} \frac{1}{n}\log_{2}M_{n}\ge R$$and $$\lim_{n\to\infty}P_{e}(\mathcal{C}_{n})=0$$