Consider binary random variables $X$ and $V$ with marginal distributions $p$ and $\pi$ respectively and also the conditional distribution $p(X=x\mid V=v)=q(x\mid v)$, where $x\in\{-b,b\}$ and $v\in\{-1,1\}$.
I am looking for a random transformation (channel) of $X$, with output $Z$, which maximizes the mutual information $I(X; Z)$ while keeping $I(V;Z)\leq \epsilon$.