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

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

\(2) Assume that $m \equiv 0 \pmod 3$. By Lemma \[lem:poly. zero 1\] (1), if the coefficients of $x^{9m+1-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-1}$ are equal to $0$ for $0 \leq i \leq 3m$, then $m=3i$. Hence the coefficients of $x^{8m+5}y^{4m+3}$ in $W_{C,f}(x,y) $ are equal to $0$. Therefore, $D_{4m+3}^{12m+8}$ is a $\{1,3,4 \}$-design. If $w \neq 4m+3$, then all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design.

\(3) Let $C'$ 