\}$.

2.  $D_{3m+3}^{12m+8}$ is never a $4$-design.

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (1), $D_{3m+3}^{12m+8}$ is a $3$-design if $t \geq 2$. If $D_{3m+3}^{12m+8}$ is a $3$-design, by Lemma \[lem: divisible\], $$\lambda_{2}=\frac{3m+2}{12m+7}3\binom{4m+2}{m}\ {\rm and}\ 
\lambda_{3}=\frac{(3m+2)(3m+1)}{(12m+7)(12m+6)}3\binom{4m+2}{m}$$ are positive integers. By a computation for $m \leq 77$, if $\lambda_{2}$ and $\lambda_{3}$ are positive integers, then we have $m \in \{ 14,37,42,63 \}$.

\(2) We have checked that $$\lambda_{4}=\frac{(3m+2)(3m+1)3m}{(12m+7)(12m+6)(12m+5)}3\binom{4m+2}{m}$$ is not a positive integer for $m \in \{ 14,37,42,63 \}$. Therefore, by Lemma \[lem: divisible\], $D_{3m+3}^{12m+8}$ is never a $4$-design.

We have the following proposition.

\[prop:type3 12m+8 harm\] Let $D_{w}^{12m+8}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+8$.

1.  If $m \not\equiv 0 \pmod 3$, all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design.

2.  Assume that $m \equiv 0 \pmod 3$. 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. In the case $w=4m+3$, $D_{4m+3}^{12m+8}$ is a $\{1,3,4 \}$-design.

3.  In the case $m=63$ and $w=255$, $D_{255}^{764}$ is not a $7$-design.

Let $C$ be an extremal Type III $[12m+8,6m+4,3m+3]$ code. Let us assume that $t=4$. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) 