= Solution
For a <discrete memoryless channel> with <transition probabilities> $M(y\mid x)$, the <conditional probability> of a length-$n$ output given a length-$n$ input factors as
$$
\mathbb P(Y_1=y_1,\ldots,Y_n=y_n\mid X_1=x_1,\ldots,X_n=x_n)=\prod_{j=1}^n M(y_j\mid x_j).
$$
The same <transition probabilities> apply at each use. Its <channel capacity> is
$$
\boxed{C=\max_{p_X}I(X;Y)=\max_{p_X}\sum_{x,y}p_X(x)M(y\mid x)\log_2\frac{M(y\mid x)}{\sum_{x'}p_X(x')M(y\mid x')} .}
$$
The unit is <bits> per channel use. For <finite alphabets> the maximum exists by <continuity> on the <probability simplex>; with more general <alphabets> one uses a <supremum> when appropriate.
An $(m,n)$ <channel code> consists of an <encoder> $c:\mathcal V_m\to\mathcal X^n$ and a <decoder> $d:\mathcal Y^n\to\mathcal V_m$. It represents each of $m$ messages by $n$ input symbols and recovers a message from the $n$ observed output symbols. Its <code rate> is \b[$R=(\log_2m)/n$ <bits> per use]. For each message $w$, its conditional error <probability> is $\mathbb P(d(Y^n)\ne w\mid W=w)$; an average error also specifies the message <probability distribution>, while the maximum error takes the largest conditional error.
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