Definition (Block Codes for the Gaussian Channel)

Given positive integers $n$ and $M=M_{n}$, a block (fixed-length) code $\mathcal{C}_{n}=(n,M)$ for a AWGN Channel with

• Average Power Constraint: $P$
• Codeword Length: $n$
• Number of Codewords: $M$
• Rate: $\frac{1}{n}\log_{2}M$ (bits per channel symbol)
• Message Set: $\mathcal{M}=\{1,2,\dots,M\}$ of $M$ information messages intended for transmission (note: messages are uniformly distributed over $\mathcal{M}$)
• Encoding Function: $$f:\mathcal{M}\to\mathbb{R}^{n}$$yielding codewords $c_{1}=f(1),\ldots,c_{M}=f(M)$ such that each codeword $c_{m}=(c_{m_{1}},\ldots,c_{m_{n}})$ of length $n$ satisfies the average power constraint $P$ $$\frac{1}{n}\sum\limits_{i=1}^{n}c_{m_{i}}^{2}\le P, \ m=1,\ldots,M$$
• Decoding function $$g:\mathbb{R}^{n}\to\mathcal{M}$$