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 $n$.


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


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? 



How did they come up with the MRRW bound?
I don't know what they were thinking in the past, but there are more modern points of view these days.


1. We can view the Delsarte LP as the symmetrization of a very large LP (or as the symmetrization of the [Lovász theta function](https://en.wikipedia.org/wiki/Lov%C3%A1sz_number) SDP on the hypercube graph with edges between strings of distance $\le d$). The Kravchuk polynomials arise naturally in this way since they are the symmetrization of the Boolean Fourier characters. Symmetrization makes the program exponentially smaller, but it does not change the value.
2. The candidate solution to the Delsarte LP has the form $(a-x) F(x)^2$ after applying Christoffel-Darboux to the candidate $\beta(x)$ in the question. The term $(a-x)$ enforces the sign constraints of the LP, and $\widehat{F}(x)$ is chosen to be an approximate eigenfunction to the matrix $a\cdot\text{Id} - A$. Here $A$ is the adjacency matrix of the standard Boolean hypercube, then symmetry-reduced into a matrix on $\{0,1, \dots, n\}$.
The approximate eigenfunction condition guarantees that the MacWilliams identities are satisfied.


See "An Elementary Proof of the First LP Bound on the Rate of Binary Codes" by Nati Linial and Elyassaf Loyfer for more on this perspective (for disclosure, they are my collaborators)


