Abstract
Three essentially different approaches to the constructive part of theÌý channel Ìý coding Ìýtheorem have been proposed by Shannon, Feinstein and Gallager, respectively, leading to upperÌý bounds Ìýon the minimal error probability achievable with a given rate and blocklength. Here,Ìý new ÌýupperÌý bounds Ìýare given on both average and maximal error probability, which are tighter than existingÌý bounds Ìýfor many ranges of blocklength andÌý channel Ìýparameters of interest. Along with converseÌý bounds , theÌý new achievability Ìý bounds Ìýallow to approximate tightly the maximum rate achievable for a given blocklength and error probability for blocklengths as short as n = 200 for both the BSC and the BEC.