Delsarte linear programming bound

Remarks

The key fact is that $\sum_{x, y \in C} G_k(x \cdot y) \ge 0$ for every $k$, because $G_k(x \cdot y)$ is a positive definite kernel on the sphere, a consequence of the addition theorem for spherical harmonics. Then $\sum_{x, y \in C} f(x \cdot y) = \sum_k f_k \sum_{x, y} G_k(x \cdot y) \ge f_0 \abs{C}^2$, since the term $k = 0$ equals $f_0 \abs{C}^2$. On the other side, the diagonal terms give $\abs{C} f(1)$ and the off-diagonal terms are at most $0$ by clause 2. So $f_0 \abs{C}^2 \le \abs{C} f(1)$. The bound was introduced by Philippe Delsarte for codes in Hamming space, and by Delsarte, Goethals and Seidel for spherical codes in 1977. With $s = 1/2$ it bounds the kissing number; Odlyzko and Sloane, and independently Levenshtein, found polynomials giving $\tau(8) \le 240$ and $\tau(24) \le 196560$, which are sharp.

Going deeper

For a student

Finding the best polynomial of a given degree is a linear program, hence the name. The method gives exact answers in dimensions $8$ and $24$, where the extremal codes are the shortest vectors of the E8 lattice and of the Leech lattice. In dimension $3$ it gives only $\tau(3) \le 13$. Cohn and Elkies adapted the idea to sphere packings in $\R^n$, which led to the solution in dimensions 8 and 24.