  We investigate universal bounds on spherical codes and spherical designs that
could be obtained using Delsarte's linear programming methods. We give a lower
estimate for the LP upper bound on codes, and an upper estimate for the LP
lower bound on designs. Specifically, when the distance of the code is fixed
and the dimension goes to infinity, the LP upper bound on codes is at least as
large as the average of the best known upper and lower bounds. When the
dimension n of the design is fixed, and the strength k goes to infinity, the LP
bound on designs turns out, in conjunction with known lower bounds, to be
proportional to k^{n-1}.
