Empirical Properties of Good Channel Codes

In this article, we revisit the classical problem of channel coding and obtain novel results on properties of capacity- achieving codes. Specifically, we give a linear algebraic characterization of the set of capacity-achieving input distributions for discrete memoryless channels. This allows us to characterize the dimension of the manifold on which the capacity-achieving distributions lie. We then proceed by examining empirical properties of capacity-achieving codebooks by showing that the joint-type of k-tuples of codewords in a good code must be close to the k- fold product of the capacity-achieving input distribution. While this conforms with the intuition that all capacity-achieving codes must behave like random capacity-achieving codes, we also show that some properties of random coding ensembles do not hold for all codes. We prove this by showing that there exist pairs of communication problems such that random code ensembles simultaneously attain capacities of both problems, but certain (superposition ensembles) do not.Due to lack of space, several proofs have been omitted but can be found at https://sites.google.com/view/yihan/