The authors call a \((2s-1)\)-design with \(s\) intersection numbers extremal. Expressing the coefficients of the Delsarte polynomial for an extremal design in a suitable manner, they obtain several interesting necessary conditions on the parameters and intersection numbers of such designs. They establish that for a fixed \(\lambda\) and \(s \geq 3\), there exist at most finitely many extremal designs. They also show that an extremal design with \(s\) intersection numbers is the Witt 5-(24,8,1) design if and only if \(s \geq 3\) and the sum of the intersection numbers is at most \(s(s-1)\).