Three essentially different approaches to the constructive part of the  channel   coding  theorem have been proposed by Shannon, Feinstein and Gallager, respectively, leading to upper  bounds  on the minimal error probability achievable with a given rate and blocklength. Here,  new  upper  bounds  are given on both average and maximal error probability, which are tighter than existing  bounds  for many ranges of blocklength and  channel  parameters of interest. Along with converse  bounds , the  new achievability   bounds  allow 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.