Definition (Fixed-Length Codes for Discrete Channels)

Given positive integers $n$ and $M$, an $(n,M)$ fixed-length code for a discrete channel $(\mathcal{X},\mathcal{Y}, \{P_{Y^{n}|X^{n}}\}_{n=1}^{\infty})$ with code length $n$ and rate $R_{n}:=\frac{1}{n}\log_{2}M$ message bits/channel use consists of:

• a message set $\mathcal{M}=\{1,\cdots,M\}$ intended for transmission
• an encoding function $$f:\mathcal{M}\to\mathcal{X}^{n}$$yielding codewords $f(1),\cdots,f(M)\in\mathcal{X}^{n}$ of length $n$. The set of codewords is called the codebook, written as $$\mathcal{C}_{n}=\{f(1),\cdots,f(M)\}$$
• a decoding function $$g:\mathcal{Y}^{n}\to\mathcal{M}$$

Definition (Average probability of error)

Given a $(n,M)$ code $\mathcal{C}_{n}$, its average probability of error is given by $$\begin{align*} P_e(\mathcal{C}_n):&=P(\hat W\not=W)\\ &=\sum\limits_{w=1}^{M}P(W=w)P(g(y^{n}\not=w|W=w))\\ &=\frac{1}{M}\sum\limits^{M}_{w=1}\lambda_{w}(\mathcal{C}_{n}) \end{align*}$$where $$\begin{align*} \lambda_{w}(\mathcal{C}_{n}):&=P(g(y^{n})\not=w|W=w))\\ &=P(g(y^{n})\not=w|X^{n}=f(w))\\ &=\sum\limits_{y^{n}\in\mathcal{Y}^{n}:g(y^{n}\not=w)}P_{Y^{n}|X^{n}}(y^{n}|x^{n}) \end{align*}$$is the code’s conditional Probability of Decoding error given that the message $w$ is sent over the channel.