For nonnegative integers n, d, and w, let A(n,d,w) be the maximum size of a code C⊆F 2 n with a constant weight w and minimum distance at least d. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on A(n,d,w). The new upper bounds imply that A(22,8,10)=616 and A(22,8,11)=672. Lower bounds on A(22,8,10) and A(22,8,11) are obtained from the (n,d)=(22,7) shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant-weight w codes of sizes A(22,8,w).