are $8$-designs simultaneously, or none of $D_{w}^{6m}$ is an $8$-design.

Let $C$ be an extremal Type IV $[6m,3m,2m+2]$ code. Let us assume that $t=8$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{8} Z_{C,f} (x,y)$, 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: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-8} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-3$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{8} h_{2} h_{6}^{m-3}   \\
& =c(f) (xy)^{8} y^{2m-6} (x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-6}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (1), the coefficients of $x^{4m-10-2i}y^{2i}$ in $(x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-6}$ are not equal to $0$ for $0 \leq i \leq 2m-5$ since $2m-6 \neq 4i-1$. Therefore, all $D_{w}^{6m}$ are $8$-designs simultaneously, or none of $D_{w}^{6m}$ is an $8$-design.

By Proposition \[prop:type4 6m not 8-design\] and \[prop:type4 6m harm\], we obtain the following theorem.

\[thm:main thm 6m\]

If $D_{w}^{6m}$ becomes a $7$-design for any $w$, then $m$ must be in the set $\{10,15 \}$.

$D_{w}^{6m}$ is never an $8$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound 