for n=6m}
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In this section, we consider the case of extremal Type IV $[6m,3m,2m+2]$ codes ($3 \leq m \leq 16$). Let $C$ be an extremal Type IV $[6m,3m,2m+2]$ code, and $D_{2m+2}^{6m}$ be the support (with duplicates omitted) design of the minimum weight of $C$. By [@MOSW1978 Theorem 18], $D_{2m+2}^{6m}$ is a $$5\mbox{-}\left(6m,2m+2,\binom{3m-3}{m-2} \right)\mbox{ design}.$$

\[prop:type4 6m not 8-design\] Let $D_{2m+2}^{6m}$ be the support $t$-design of the minimum weight of an extremal Type IV code of length $n=6m$.

1.  If $t \geq 6$, then $D_{2m+2}^{6m}$ is a $7$-design and $m$ must be in the set $\{10,15 \}$.

2.  $D_{2m+2}^{6m}$ is never an $8$-design.

\(1) If $D_{2m+2}^{6m}$ is a $7$-design, by Lemma \[lem: divisible\], $$\lambda_{6}=\frac{2m-3}{6m-5}\binom{3m-3}{m-2} \mbox{ and } 
\lambda_{7}=\frac{(2m-3)(2m-4)}{(6m-5)(6m-6)}\binom{3m-3}{m-2}$$ are positive integers. By a computation for $m \leq 16$, if $\lambda_{6}$ and $\lambda_{7}$ are positive integers, we have $m \in \{10,15 \}$.

\(2) We have checked that $$\lambda_{8}=\frac{(2m-3)(2m-4)(2m-5)}{(6m-5)(6m-6)(6m-7)}\binom{3m-3}{m-2}$$ is not a positive integer for $m \in \{10,15 \}$. Therefore, by Lemma \[lem: divisible\], $D_{2m+2}^{6m}$ is never an $8$-design.

For $t \geq 8$, we give the following proposition.

\[prop:type4 6m harm\] Let $D_{w}^{6m}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m$. Then all $D_{w}^{6m}$ 