Refinement of the random coding bound

An improved pre-factor for the random coding bound is proved. Specifically, for channels with critical rate not equal to capacity, if a regularity condition is satisfied (resp. not satisfied), then for any $\epsilon >0$ a pre-factor of $O(N^{-\frac{1}{2}\left( 1 - \epsilon + \bar{\rho}^\ast_R \right)})$ (resp. $O(N^{-\frac{1}{2}})$) is achievable for rates above the critical rate, where $N$ and $R$ is the blocklength and rate, respectively. The extra term $\bar{\rho}^\ast_R$ is related to the slope of the random coding exponent. Further, the relation of these bounds with the authors' recent refinement of the sphere-packing bound, as well as the pre-factor for the random coding bound below the critical rate, is discussed.