How quickly can we approach channel capacity?

Recent progress in code design has made it crucial to understand how quickly communication systems can approach their limits. To address this issue for the channel capacity C, we define the nonasymptotic capacity C_NA(n, ε) as the maximal rate of codebooks that achieve a probability ε of codeword error while using codewords of length n. We prove for the binary symmetric channel that C_NA(n,ε)=C-K(ε)/√n+o(1/√n), where K(ε) is available in closed form. We also describe similar results for the Gaussian channel. These results may lead to more efficient resource usage in practical communication systems.