of thm1.3}
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Case for $n=12m$ {#sec: case for n=12m}
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In this section, we consider the case of extremal Type III $[12m,6m,3m+3]$ codes ($m \leq 69$). Let $C$ be an extremal Type III $[12m,6m,3m+3]$ code, $D_{3m+3}^{12m}$ 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 $$2\binom{12m}{5}\binom{4m-2}{m-1}\left/\binom{3m+3}{5}.\right.$$ Therefore, by the Assmus–Mattson theorem, $D_{3m+3}^{12m}$ is a $5$-design with parameters $$\left(12m,3m+3,\binom{4m-2}{m-1} \right).$$

\[prop:type3 12m not 8-design\]

1.  If $t \geq 6$, then $D_{3m+3}^{12m}$ is a $7$-design and $m$ must be in the set $\{15,38,43,64 \}$.

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

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (1), $D_{3m+3}^{12m}$ is a $7$-design if $t \geq 6$. If $D_{3m+3}^{12m}$ is a $7$-design, by Lemma \[lem: divisible\], $$\lambda_{6}=\frac{3m-2}{12m-5}\binom{4m-2}{m-1}\ {\rm and}\ 
\lambda_{7}=\frac{(3m-2)(3m-3)}{(12m-5)(12m-6)}\binom{4m-2}{m-1}$$ are positive integers. By a computation for $m \leq 69$, if $\lambda_{6}$ and $\lambda_{7}$ are positive integers, we have $m \in \{15,38,43,64 \}$.

\(2) We have checked that $$\lambda_{8}=\frac{(3m-2)(3m-3)(3m-4)}{(12m-5)(12m-6)(12m-7)}\binom{4m-2}{m-1}$$ is not a positive integer for $m \in \{15,38,43,64 \}$. Therefore, by Lemma \[lem: divisible\], $D_{3m+3}^{12m}$ is never an $8$-design.

For $t=8$, we give the following proposition.

\[prop:type3 12m harm\] Let $D_{w}^{12m}$ be the support 