i \leq 2m-3$ since $2m-4 \neq 4i-1$. Therefore, all $D_{w}^{6m+2}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $6$-design.

\(2) Let us assume that $t=7$. Then by the Theorem \[thm:invariant\] (2) 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 $Z_{C,f} (x,y) \in I_{G_{4},\chi_{1,1}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = q_{3}r_{3}\sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-9} h_{6}^{i}.$$ 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: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{7} q_{3}r_{3} h_{6}^{m-3}   \\
& =c(f) (xy)^{7} y^{2m-5} (x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-5}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (2), the coefficients of $x^{4m-7-2i}y^{2i}$ in $(x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-5}$ are not equal to $0$ for $0 \leq i \leq 2m-4$. Therefore, all $D_{w}^{6m+2}$ are $7$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $7$-design.

\(3) 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: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-7} h_{6}^{i}.$$ Since 