Two channels are equivalent if their maximum likelihood (ML) decoders coincide for every code. We show that this equivalence relation partitions the space of channels into a generalized hyperplane arrangement. With this, we define a coding distance between channels in terms of their ML-decoders which is meaningful from the decoding point of view, in the sense that the closer two channels are, the larger is the probability of them sharing the same ML-decoder. We give explicit formulas for these probabilities.
@cite_12: This paper studies the basic question of whether a given channel @math can be dominated (in the precise sense of being more noisy) by a @math -ary symmetric channel. The concept of "less noisy" relation between channels originated in network information theory (broadcast channels) and is defined in terms of mutual information or Kullback-Leibler divergence. We provide an equivalent characterization in terms of @math -divergence. Furthermore, we develop a simple criterion for domination by a @math -ary symmetric channel in terms of the minimum entry of the stochastic matrix defining the channel @math . The criterion is strengthened for the special case of additive noise channels over finite Abelian groups. Finally, it is shown that domination by a symmetric channel implies (via comparison of Dirichlet forms) a logarithmic Sobolev inequality for the original channel.
@cite_2: A partial ordering is defined for discrete memoryless channels. It is transitive and is preserved under channel operations of addition and multiplication. The main result proved is that if K 1 and K 2 are such channels, and K 1 ⊇ K 2 , then if a code exists for K 2 , there exists at least as good a code for K 1 , in the sense of probability of error.
Our approach to study the geometry of the space of channels has an intersection with the concept of , as introduced by Shannon @cite_2 and as presented, for example, by Makur and Polyanskiy @cite_12 . Using the notation of Makur and Polyanskiy, given two channels with transition matrices @math and @math of size @math and @math respectively, with @math and @math , one says that @math @math if there are two families @math and @math of channels (with @math being an @math transition matrix and @math an @math transition matrix) and probability mass function @math over the set @math such that [ Q= k=1 ^ m g ( k ) B_ k PA_ k . ]