I have been studying coding theory and a bit of information theory as well, and found this problem:
Given the channel with channel matrix
\begin{bmatrix} 1/6 & 1/3 & 1/2\\ 1/3 & 1/2 & 1/6\\ 1/2 & 1/6 & 1/3 \end{bmatrix} and the following input distribution $$p(x=0)=0.5,\quad p(x=1)=p(x=2)=0.25$$ find the best decision scheme (for transmission with no coding) and the associated average and maximum probabilities of error.
The associated average of error is given by $$p_e:=\sum_{c\in C}p(\text{error}|c)p(c)=\sum_{c\in C}\sum_{u\notin B_c}p(u|c)p(c),$$ where $B_c=\{u\in \mathbb{F}_q^n:f(u)=c\}$ and $f:\mathbb{F}_q^n\to C\cup\{*\}$ is the decision scheme ($f(u)=*$ iff $u$ is not decoded).
From a decoder's point of view, we have $$p_e=\sum_{u\in\mathbb{F}^n_q}p(\text{error})p(u)=1-\sum_{u\in\mathbb{F}_q^n}p(f(u)|u)p(u).$$
And so my goal is to maximize $p(f(u)| u) $ for each $u$. But I'm having some trouble on how to do it and I haven't found much material on this. Maybe someone could help me? Thanks in advance.