D_{w} \mbox{ is a } t\mbox{-design}\}.\end{aligned}$$ Note that $\delta(C) \leq s(C)$. In our previous papers , we considered the following problems.

\[problem:1\] Find an upper bound of $s(C)$.

\[problem:2\] Does the case where $\delta(C) < s(C)$ occur?

For Problem \[problem:1\], there is no known example of a $6$-design obtained from the Assmus–Mattson theorem. For Problem \[problem:2\], if $C$ is an extremal Type II code, there is no known example of $\delta(C)<s(C)$ [@extremal; @design2; @M-N]. In [@support; @design; @triply; @even; @code; @48; @M-N], we gave examples of $\delta(C)<s(C)$ for non self-dual codes.

In the present paper, we consider extremal Type III and IV codes. Let $C$ be an extremal Type III or IV code of length $n$. In 1999, it was shown by Zhang [@Zhang(1999)] that $C$ does not exist if for Type III and $$n=
\left\{
\begin{array}{l}
12m\ (m \geq 70), \\
12m+4\ (m \geq 75), \\
12m+8\ (m \geq 78),
\end{array}
\right.$$ for Type IV and $$n=
\left\{
\begin{array}{l}
6m\ (m \geq 17), \\
6m+2\ (m \geq 20), \\
6m+4\ (m \geq 22). 
\end{array}
\right.$$

The main results of the present paper are the following theorems.

\[thm:main upper bound III\] Let $C$ be an extremal Type III code of 