Maximizing entropy on a channel
Charles E.Radke, "Necessary and sufficient conditions on conditional probabilities to maximize entropy", Information and Control 9, 279-284 (1966), https://doi.org/10.1016/S0019-9958(66)90165-3 gives a proof of Shannon's Theorem 8:


  A necessary and sufficient condition such that $\eta_c$ is maximized and equal to $1$ is that the set of symbol lengths ${l_{ij}}^v$ must satisfy:
  $${p_{ij}}^v = (\beta_j/\beta_i)\omega_r^{-{l_{ij}}^v},\tag{3}$$
  where $\beta_i$ and $\beta_j$ are respectively the $(k, i)$th and $(k, j)$th cofactor of the matrix $A$ for any $k$.


The matrix $A$ is defined earlier as:


  $$A \triangleq \left[\textstyle\sum\omega^{-{l_{ij}}^v} - \delta_{ij}\right];$$


which is something you can recognize in Shannon's Theorem 1.