of an extremal Type III code of length $n=12m+4$ $($$m \leq 74$$)$.

1.  If $D_{w}^{12m+4}$ becomes a $5$-design for any $w$, then $m$ must be in the set\
    $\{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}$.

2.  If $D_{w}^{12m+4}$ becomes a $6$-design for any $w$, then $m$ must be $67$.

3.  In the case $m=67$, $D_{w}^{808}$ is a $3,5$ or $6$-design for any $w$.

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

Thus the proof of Theorem \[thm:main upper bound III\] (2) is completed.

Case for $12m+8$ {#sec: case for n=12m+8}
----------------

In this section, we consider the case of extremal Type III $[12m+8,6m+4,3m+3]$ codes ($m \leq 77$). Let $C$ be an extremal Type III $[12m+8,6m+4,3m+3]$ code, and $D_{3m+3}^{12m+8}$ be the support (with duplicates omitted) design of the minimum weight of $C$. By [@mallows-sloane Theorem 2], the number of codewords of minimum nonzero weight of $C$ is equal to $$6(12m+8)\frac{(4m+2)!}{m!(3m+3)!}.$$ Therefore, by the Assmus–Mattson theorem, $D_{3m+3}^{12m+8}$ is a $1$-design with parameters $$\left(12m+8,3m+3,3\binom{4m+2}{m} \right).$$

\[prop:type3 12m+8 not 4-design\] Let $D_{3m+3}^{12m+8}$ be the support $t$-design of the minimum weight of an extremal Type III code of length $n=12m+8$.

1.  If $t \geq 2$, then $D_{3m+3}^{12m+8}$ is a $3$-design and $m$ must be in the set $\{ 14,37,42,63 