\}$.

$D_{w}^{12m}$ is never an $8$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound III\] (1) is completed.

Case for $12m+4$ {#sec: case for n=12m+4}
----------------

In this section, we consider the case of extremal Type III $[12m+4,6m+2,3m+3]$ codes ($m \leq 74$). Let $C$ be an extremal Type III $[12m+4,6m+2,3m+3]$ code, and $D_{3m+3}^{12m+4}$ 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 $$2(12m+4)(12m+3)(12m+2)\frac{(4m)!}{m!(3m+3)!}.$$ Therefore, by the Assmus–Mattson theorem, $D_{3m+3}^{12m+4}$ is a $3$-design with parameters $$\left(12m+4,3m+3,\binom{4m}{m} \right).$$

\[prop:type3 12m+4 not 7-design\] Let $D_{3m+3}^{12m+4}$ be the support $t$-design of the minimum weight of an extremal Type III code of length $n=12m+4$.

1.  If $t \geq 4$, then $D_{3m+3}^{12m+4}$ is a $5$-design and $m$ must be in the set\
    $\{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}$.

2.  If $D_{3m+3}^{12m+4}$ is a $6$-design, $m$ must be $67$.

3.  $D_{3m+3}^{12m+4}$ is never a $7$-design.

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (1), $D_{3m+3}^{12m+4}$ is a $5$-design if $t \geq 4$. If $D_{3m+3}^{12m+4}$ is a $5$-design, by Lemma \[lem: divisible\], $$\lambda_{4}=\frac{3m}{12m+1}\binom{4m}{m} \mbox{ and } 
\lambda_{5}=\frac{3m(3m-1)}{(12m+1)12m}\binom{4m}{m}$$ are positive integers. By a computation for $m \leq 74$, if $\lambda_{4}$ and $\lambda_{5}$ are 