The Assmus–Mattson nonexistence conjecture for higher-weight codes

Let $C$ be an $ell$-weight $[n.k.d]$ code over ${\mathbb F}_{q}$ satisfying the Assmus–Mattson condition. Assume that $d^\perp\geq 2ell+1$ and

$$W_{C}(x,y)=x^{n}+\sum_{1\leq i\leq ell}\alpha_i x^{n-d_i}y^{d_i}.$$

Assmus–Mattson nonexistence conjecture. Then

$$1+\alpha_1+\alpha_2+\cdots+\alpha_{ell}=\sum_{i=0}^{ell}\binom{n}{i}(q-1)^i=q^k.$$

Moreover, if $ell\geq 4$, codes corresponding to solutions of

$$\sum_{i=0}^{ell}\binom{n}{i}(q-1)^i=q^k$$

do not exist. Consequently, $d^\perp\leq 2ell$ and $t\leq 2ell-1$ for $ell\geq 4$. The claim extends the preceding two- and three-weight observations and predicts nonexistence of the corresponding codes in all cases with at least four weights.