We consider block codes whose rate converges to the channel capacity with increasing block length at a certain speed and examine the best possible decay of the probability of error. We prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes.

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Moderate deviation behavior of coding for discretememoryless channels is investigated. That is, we consider block codes whose rate converges to the channel capacity from below with increasing block length with a certain rate and examine the best 'sub-exponential' decay in the maximal probability of error. We prove that a moderate deviation principle (M.D.P.) holds for all convergence rates between the large deviation and the central limit theorem regimes, under some mild assumptions on the channel. The rate function of the M.D.P. is explicitly characterized.

We provide a refinement of the sphere-packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the pre-factor in front of the exponential term. The order of our pre-factor is Ω(N − 1 2 (1+ǫ+ρ * R ) ) for any ǫ > 0, where ρ * R is the maximum absolute-value subdifferential of the sphere-packing exponent at rate R and N is the blocklength. 2 Haroutunian [8], [12, Theorem 2.5.3], where |X | and |Y| are the cardinalities of the input and output alphabets, respectively. (The original sphere-packing bound, derived by Shannon-Gallager-Berlekamp [7, Theorem 2], had an Θ(e − √ N ) pre-factor.) Clearly, there is a considerable gap between the orders of the pre-factors in the upper and lower bounds.Recently, the authors have been working to reduce the gap between the pre-factors. The recent paper [21] considers symmetric channels and refines the sphere-packing lower bound by proving a pre-factor of Θ(N − 1is the slope of the sphere-packing exponent at point R. The paper [24] proves a refined random coding bound with a pre-factor of O(N − 1 2 (1−ǫ+ρ * R ) ) for any ǫ > 0, for a broad class of channels, which includes all positive channels with positive dispersion. Here,ρ * R is related to the subgradient of the random coding exponent, which reduces to |E ′ r (R)| for the case of completely symmetric or positive and symmetric channels; hence the optimal order of the pre-factor is determined, up to the sub-polynomial terms.This work is a generalization of [21] to asymmetric channels. We prove a lower bound for constant composition codes with a pre-factor of Ω(N − 1 2 (1+ǫ+ρ * R ) ) for any ǫ > 0, where ρ * R is the maximum absolute-value subgradient of the sphere-packing exponent. While the essential approach is similar to that of [21], the asymmetry of the channel results in a significantly more involved argument compared to its symmetric counterpart. Although some improved finite-N bounds could be extracted from the proofs in this paper, the task of optimizing these bounds and numerically comparing them to the existing bounds is not pursued, since we focus on the asymptotic characterization.An analogy to sums of i.i.d. random variables is instructive. The small, medium, and large error probability regimes of channel coding correspond to large deviations, moderate deviations, and central limit theory of i.i.d. sums of random variables, respectively. Along the same analogy, the setup of this work resembles the exact asymptotics problem in large deviations [25], [26, Theorem 3.7.4]. This problem aims to determine the pre-factor of the exponentially vanishing term in the large deviations theorem. Bahadur and Ranga Rao [25] characterized this pre-factor, Θ(1/ √ N ), including the constant, under some regularity conditions. Their result, in the form stated by Dembo and Zeitouni [26, Theorem 3.7.4], is the following:and Λ * (·) is the Fenchel-Legendre transform of Λ(·).If X 1 is a lattice random variable, then the order of the pre-factor is the same, but the constant is different. Henc...

Abstract-We provide an improved random coding bound for a class of discrete memoryless channels that improves the prefactor in front of the exponent. Specifically, while the tightest known bounds have a constant pre-factor, the presented bound has a pre-factor of order O(1/ √ N ), where N is the blocklength.