  The sphere packing problem asks for the greatest density of a packing of
congruent balls in Euclidean space. The current best upper bound in all
sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We
revisit their argument and improve their bound by a constant factor using a
simple geometric argument, and we extend the argument to packings in hyperbolic
space, for which it gives an exponential improvement over the previously known
bounds. Additionally, we show that the Cohn-Elkies linear programming bound is
always at least as strong as the Kabatiansky-Levenshtein bound; this result is
analogous to Rodemich's theorem in coding theory. Finally, we develop
hyperbolic linear programming bounds and prove the analogue of Rodemich's
theorem there as well.
