\[lem: divisible\], $D_{2m+2}^{6m+2}$ is never an $8$-design.

For $t \geq 6$, we give the following proposition.

\[prop:type4 6m+2 harm\] Let $D_{w}^{6m+2}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m+2$.

1.  All $D_{w}^{6m+2}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $6$-design.

2.  All $D_{w}^{6m+2}$ are $7$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $7$-design.

3.  In the case $m=11$. All $D_{w}^{68}$ are $8$-designs simultaneously, or none of $D_{w}^{68}$ is an $8$-design.

Let $C$ be an extremal Type IV $[6m+2,3m+1,2m+2]$ code.

\(1) Let us assume that $t=6$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{6} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-5} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{6} h_{2} h_{6}^{m-2}   \\
& =c(f) (xy)^{6} y^{2m-4} (x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-4}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (1), the coefficients of $x^{4m-6-2i}y^{2i}$ in $(x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-4}$ are not equal to $0$ for $0 \leq 