Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is

Suppose $C \subset \mathbb{F}_2^n $ is a code such $d(C)\ge d$. Let $\beta(x) = 1+ \sum_{k=1}^{n} y_k K_k (x)$ be a polynomial such that $y_k \ge 0$ but $\beta(j) \le 0$ for $j=d, d+1,\dots ,n$. Then, we have that $|C| \le \beta(0)$.

Here $K_k(x)$ are the Kravchuk polynomials. In the proof of the MRRW bound, upto scaling, they basically come up with the following polynomial $\beta$ for a general $n$.

$$\beta(x) =\frac{1}{x-a} \left[ K_t(a) K_{t+1}(x) - K_{t+1}(a)K_{t}(x) \right]^{2}$$

After using the Christoffel-Darboux formula the values of $t$ and $a$ are adjusted to make it optimal.

There is no justification for why such a polynomial was chosen other than that it works. Is there anything more that can be said over why this polynomial was chosen?