Antipodal code coefficient conjecture under the Assmus–Mattson condition

Let $C$) be a binary antipodal $[n,k]$ code with weight enumerator

$$W_C(x,y)=x^n+\sum_{i\geq 1}\alpha_i x^{n-d_i}y^{d_i}+y^n.$$

Assume that $C$ satisfies the Assmus–Mattson condition with $d^\perp-t=\ell$, where $\ell\geq 6$, and that $d^\perp\geq 2\ell+2$.

Antipodal code coefficient conjecture. One has

$$\begin{cases} \displaystyle 1+\alpha_1+\cdots+\alpha_{\ell/2}=\sum_{i=0}^{\ell}\binom{n-1}{i}=2^{k-1} & \text{if } \ell\equiv 0\pmod{2},\\\\ \displaystyle 1+\alpha_1+\cdots+\frac{\alpha_{\lceil\ell/2\rceil}}{2}=\sum_{i=0}^{\ell}\binom{n-1}{i}=2^{k-1} & \text{if } \ell\equiv 1\pmod{2}, \end{cases}$$

Moreover, $t\leq\ell$.

This conjecture proposes a general relation between the weight-enumerator coefficients of antipodal binary codes and the Assmus–Mattson parameters when $\ell\geq 6$; the preceding cases $d^\perp-t=4$ and $d^\perp-t=5$ motivate it, while the asserted identities and the bound on $t$ remain to be established in the stated generality.