You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: 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.

[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 [2] and as presented, for example, by Makur and Polyanskiy [1] . 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 . ]