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 [1]. Kabatiansky and Levenshtein [2] 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. In this work 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.