Best upper bound on rate for q-ary codes
This is not the best bound even for $q=2$; in fact, this is not the best bound derived from the Delsarte linear program; see the paper "On the optimum of Delsarte's linear program" by Samorodnitsky (1998). 

Thus, a better analysis of the linear program is likely to improve the bounds over larger $q$.  Even for $q=2$, this is a complicated analysis, so I don't know whether anybody has performed it for larger $q$.