A New Upper Bound on the Reliability Function of the Gaussian Channel

We derive a new upper bound on the exponent of error probability for the best possible codes in the Gaussian channel. This bound is tighter than the known upper bounds (the sphere-packing and minimum-distance bounds proved in Shannon's classical 1959 paper and their low-rate improvement by Kabatiansky and Levenshtein). The proof is accomplished by studying asymptotic properties of codes on the Euclidean n-dimensional sphere. First we prove that the distance distribution of codes of large size necessarily contains a large component. A general theorem establishing the estimate is proved simultaneously for codes on the Euclidean sphere and in real and complex projective spaces.