length $n$.

1.  Assume that $n=12m$.

    1.  If $m \notin \{15,38,43,64\}$, $\delta(C)=s(C)=5$.

    2.  If $m \in \{15,38,43,64\}$, $\delta(C)=s(C)=5$ or $7$.

2.  Assume that $n=12m+4$.

    1.  If $m \notin \{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}$, $\delta(C)=s(C)=3$.

    2.  If $m \in \{11,18,21,25,32,39,43,46,49,54,60,65,68,74 \}$, $\delta(C)=s(C)=3$ or $5$.

    3.  If $m=67$, $\delta(C)=s(C)=3,5$ or $6$.

3.  Assume that $n=12m+8$.

    1.  If $m \notin \{ 14,37,42,63 \}$, $\delta(C)=s(C)=1$.

    2.  If $m \in \{ 14,37,42 \}$, $\delta(C)=s(C)=1$ or $3$.

    3.  If $m=63$, $\delta(C)=s(C)=1$ or $\delta(C)=3$, $4 \leq s(C) \leq 6$.

\[thm:main upper bound IV\] Let $C$ be an extremal Type IV code of length $n$.

1.  Assume that $n=6m$ $(m \neq 1,2)$.

    1.  If $m \notin \{10,15\}$, $\delta(C)=s(C)=5$.

    2.  If $m \in \{10,15\}$, $\delta(C)=s(C)=5$ or $7$.

2.  Assume that $n=6m+2$.

    1.  If $m \neq 11$, $\delta(C)=s(C)=3$.

    2.  If $m=11$, $\delta(C)=s(C)=3, 5, 6$ or $7$.

3.  Assume that $n=6m+4$.

    1.  If $m \in \{1,2,4,13\}$, $\delta(C)=s(C)=1$.

    2.  If $m\in \{3,5,6,7,8,10,11,12,15,16,17,18,20,21\}$, $\delta(C)=s(C)=1$ or $3$.

    3.  If $m=9$, $\delta(C)=s(C)=1,3$ or $4$.

    4.  If $m \in \{ 14,19 \}$, $\delta(C)=s(C)=1,3,4$ or $5$.

For Problem \[problem:1\], we conclude that $s(C) \leq 7$ for any extremal Type III or IV code $C$. For Problem \[problem:2\], we have the following proposition.

Let $C$ be an extremal Type III or 