$t$-design of weight $w$ of an extremal Type III code of length $n=12m$. Then all $D_{w}^{12m}$ are $8$-designs simultaneously, or none of $D_{w}^{12m}$ is an $8$-design.

Let us assume that $t=8$, and $C$ is an extremal Type III $[12m,6m,3m+3]$ code. Then by the Theorem \[thm:invariant\] (1) 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 $Z_{C,f} (x,y) \in I_{G_{3},\chi_{0,1}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = p_{4} \sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-5} g_{12}^{i}.$$ Since the minimum weight of $C$ is $3m+3$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{8} p_{4}g_{4} g_{12}^{m-2}   \\
& =c(f) (xy)^{8} y^{3m-5} (x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-5}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (1), the coefficients of $x^{9m-11-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-5}$ are not equal to $0$ for $0 \leq i \leq 3m-4$ since $3m-5 \neq 9i-1$. Therefore, all $D_{w}^{12m}$ are $8$-designs simultaneously, or none of $D_{w}^{12m}$ is an $8$-design.

By Proposition \[prop:type3 12m not 8-design\] and \[prop:type3 12m harm\], we obtain the following theorem.

\[thm:main thm 12m\]

If $D_{w}^{12m}$ becomes a $7$-design for any $w$, then $m$ must be in the set $\{15,38,43,64 