§8.4, p. 160 — the distance distribution $\tilde x(C)$ sums to $|C|$ and is feasible for the Delsarte LP

For every code $C \subseteq \{0,1\}^n$, the quantities $\tilde x_i(C) = \frac{1}{|C|}|\{(\mathbf w,\mathbf w') \in C^2 : d_H(\mathbf w,\mathbf w') = i\}|$ satisfy

and whenever $C$ is a nonempty code with distance $d$, the vector $(\tilde x_0,\dots,\tilde x_n)$ is a feasible solution of the Delsarte linear program: $\tilde x_0 = 1$, $\tilde x_i = 0$ for $i = 1,\dots,d-1$, $\sum_{i=0}^n K_t(n,i)\tilde x_i \ge 0$ for $t = 1,\dots,n$, and $\tilde x_i \ge 0$ for all $i$.

This is the step that turns a code into a feasible solution of the linear program, from which the Delsarte bound follows.

Formalization Note The book's $\tilde x_i$ divides by $|C|$ and its claim $\tilde x_0 = 1$ presupposes $C \ne \emptyset$; the hypothesis C.Nonempty makes this explicit. The identity $\sum_i \tilde x_i = |C|$ holds (trivially) for the empty code as well and is stated under the same hypothesis.