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abstract: 'Let $C$ be an extremal Type III or IV code and $D_{w}$ be the support design of $C$ for a weight $w$. We introduce the two numbers $\delta(C)$ and $s(C)$: $\delta(C)$ is the largest integer $t$ such that, for all wight, $D_{w}$ is a $t$-design; $s(C)$ denotes the largest integer $t$ such that there exists a $w$ such that $D_{w}$ is a $t$-design. In the present paper, we consider the possible values of $\delta(C)$ and $s(C)$.'
author:
- 'Tsuyoshi Miezaki[^1] and Hiroyuki Nakasora[^2]'
title: 'An upper bound of the value of $t$ of the support $t$-designs of extremal Type III and IV codes'
---

#### Keywords:

self-dual codes, $t$-designs, Assmus–Mattson theorem, harmonic weight enumerators.

#### 2010 MSC:

Primary 94B05; Secondary 05B05.

Introduction
============

Let $D_{w}$ be the support design of a code $C$ for a weight $w$. Then it is known from the Assmus–Mattson theorem [@assmus-mattson] that, if $C$ is an extremal Type III (resp. Type IV) code, for all $w$, $D_{w}$ is a $5$-, $3$- and $1$-design for $n=12m$ (resp. $n=6m$), $12m+4$ (resp. $n=6m+2$) and $12m+8$ (resp. $n=6m+4$), respectively.

Let $$\begin{aligned}
\delta(C)&:=\max\{t\in \mathbb{N}\mid \forall w, 
D_{w} \mbox{ is a } t\mbox{-design}\},\\ 
s(C)&:=\max\{t\in \mathbb{N}\mid \exists w \mbox{ s.t.~} 
