The Rate of Approach to Ideal CodingProceedings Institute of Radio Engineers, volume 43, 1955.

Let C be the capacity of a noisy discrete channel without memory. Consider codes for the channel consisting of 2 ^Hn sequences, each sequence being n symbols long. The basic coding theorem for a noisy channel states that by taking n sufficiently large it is possible to find codes such that both ¿ = C - H and the probability of error after reception, PE , are arbitrarily small. This paper is concerned with the problem of estimating the necessary value of n as a function of PE and ¿. Both upper and lower bounds are found in fairly general cases. The upper and lower bounds for n approach each other percentage-wise as ¿. and PE approach zero, giving an asymptotic formula for n. Various special cases are investigated in detail, such as the symmetric binary channel and the flat gaussian channel.