be an extremal Type III $[764,382,192]$ code and $C'_{255}$ be the set of codewords of weight 255 of $C'$. By [@mallows-sloane], we have $$\begin{aligned}
|C'_{255}|=& 57722041604247479907056082274041845325097239194558706847\\
& 5581740325339465514100889807420805771366809484288.\end{aligned}$$ Let $D_{255}^{764}$ be a $t$-$(764,255,\lambda_{t})$ design. Then we have $$\lambda_{t}=\frac{255\cdot254\cdots(255-t+1)}{764\cdot763\cdots(764-t+1)}|C'_{255}|.$$ We have checked that $\lambda_{t}$ is a positive integer for $t=1,2,\ldots, 6$ and $\lambda_{7}$ is not a positive integer. Thus $D_{255}^{764}$ is not a $7$-design.

By Proposition \[prop:type3 12m+8 not 4-design\] and \[prop:type3 12m+8 harm\], we obtain the following theorem.

\[thm:main thm 12m+8\] Let $D_{w}^{12m+8}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+8$ $($$m \leq 77$$)$.

1.  If $D_{w}^{12m+8}$ becomes a $3$-design for any $w$, then $m$ must be in the set $\{ 14,37,42,63 \}$.

2.  In the case $m \in \{ 14,37,42 \}$, $D_{w}^{12m+8}$ is a $1$ or $3$-design for any $w$.

3.  Assume that $m=63$. If $w \neq 255$, $D_{w}^{764}$ is a $1$ or $3$-design and $D_{255}^{764}$ is a $1, 4, 5$ or $6$-design.

4.  $D_{w}^{12m+8}$ is never a $7$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound III\] (3) is completed.

Proof of Theorem \[thm:main upper bound IV\] {#sec: proof of thm1.4}
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Case for $n=6m$ {#sec: case 