(xy)^{6} y^{3m-3} (x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-3}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (1), the coefficients of $x^{9m-5-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-3}$ are not equal to $0$ for $0 \leq i \leq 3m-2$ since $3m-3 \neq 9i-1$. Therefore, all $D_{w}^{12m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $6$-design.

\(2) Let us assume that $t=7$. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) (xy)^{7} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{3},\chi_{1,2}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = p_{4}^{2}p_{6} \sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-6} g_{12}^{i}.$$ 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: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{7}p_{4}^{2}p_{6} g_{12}^{m-2}  \\
& =c(f) (xy)^{7} y^{3m-4} (x^{6}-20x^{3}y^{3}-8y^{6}) (x^{3}-y^{3})^{3m-4}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (2), the coefficients of $x^{9m-6-3i}y^{3i}$ in $(x^{6}-20x^{3}y^{3}-8y^{6}) (x^{3}-y^{3})^{3m-4}$ are not equal to $0$ for $0 \leq i \leq 3m-2$. Therefore, all $D_{w}^{12m+4}$ are $7$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $7$-design.

By Proposition \[prop:type3 12m+4 not 7-design\] and \[prop:type3 12m+4 harm\], we obtain the following theorem.

\[thm:main thm 12m+4\] Let $D_{w}^{12m+4}$ be the support $t$-design of weight $w$ 