A new upper bound on the reliability function of the Gaussian channel

Upper bounds on the reliability function of the Gaussian channel were derived by Shannon in 1959. Kabatiansky and Levenshtein (1978) obtained a low-rate improvement of Shannon´s “minimum-distance bound”. Together with the straight-line bound this provided an improvement upon the sphere-packing bound in a certain range of code rate. We prove a bound better than the KL bound on the reliability function. Employing the straight-line bound, we obtain a further improvement of Shannon´s results. As intermediate results we prove lower bounds on the distance distribution of spherical codes and a tight bound on the exponent of Jacobi polynomials of growing degree in the entire orthogonality segment