This is much more lucidly stated as follows: $${}$$ We have $N$ sources $X=[X_1,\dots,X_N]^T$ where the $X_n$ are independent and $X_n\sim B(\alpha_n)$. We use a binary linear transformation $B:\{0,1\}^N\to\{0,1\}^K$ to form an input $Y:=BX$. We put $Y$ through $K$ independent binary symmetric channels $P(Z_k|Y_k)$ each with parameter $\beta_k$, to produce an output $Z=[Z_1,\dots,Z_K]^T$. What choice of $B$ maximizes $I(X;Z)?$

It came up when trying to find the information capacity of a situation where one transmitter is broadcasting to a collection of receivers that can't see each other's receptions directly, but can only conference a certain amount of information to all the others.