positive integers, then we have $$m \in \{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}.$$

(2)If $D_{3m+3}^{12m+4}$ is a $6$-design, by Lemma \[lem: divisible\], $$\lambda_{6}=\frac{3m(3m-1)(3m-2)}{(12m+1)12m(12m-1)}\binom{4m}{m}$$ is a positive integer. Then we have $m=67$.

\(3) We have checked that $$\lambda_{7}=\frac{3m(3m-1)(3m-2)(3m-3)}{(12m+1)12m(12m-1)(12m-2)}\binom{4m}{m}$$ is not a positive integer for $m=67$. Therefore, by Lemma \[lem: divisible\], $D_{3m+3}^{12m+4}$ is never a $7$-design.

For $t \geq 6$, we give the following proposition.

\[prop:type3 12m+4 harm\] Let $D_{w}^{12m+4}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+4$.

1.  All $D_{w}^{12m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $6$-design.

2.  All $D_{w}^{12m+4}$ are $7$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $7$-design.

Let $C$ be an extremal Type III $[12m+4,6m+2,3m+3]$ code.

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