Is maximum entropy noise the worst?

We find the worst constrained noise processes for the additive noise channel. If the signal and noise covariances lie in bounded convex sets, we show that a random Gaussian codebook and a decoding scheme based on a Gaussian metric achieve the minimax rate. We demonstrate the solution to the game-theoretic problem when we have a banded matrix constraint (specified up to a certain covariance lag) on the noise covariance matrix and show that under certain conditions (sufficient input power) the maximum entropy noise is also the solution to the minimax problem.