On the capacity of the Gaussian channel with a finite number of input levels

An analytic confirmation of an observation made by G. Ungerboeck (1965) that approximately optimal performance on a Gaussian channel with capacity C can be obtained with about 2^C+1 levels is given. In particular, the results imply that by using about 2^C-1 levels, a rate of nearly C-1 bit can be achieved, and that by using 2^C+1 levels a rate of about C-0.4 bits can be achieved