Posterior matching variants and fixed-point elimination

The posterior matching scheme is known to achieve capacity for a large class of memoryless channels with noiseless feedback. In this contribution, it is shown that whenever the posterior matching kernel admits a fixed point, the corresponding scheme is not ergodic and cannot achieve any positive rate. The source of the problem is traced back to the input ordering implicit in the definition of the scheme. It is then shown how for any discrete memoryless channel, a simple (and easily computable) input permutation eliminates the fixed point phenomena and allows a corresponding variant of the scheme to achieve capacity. This notion is then systematically extended to the case of continuous alphabet memoryless channels.