---
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 $$\begin{aligned}
\delta(C)&:=\max\{t\in \mathbb{N}\mid \forall w, 
D_{w} \mbox{ is a } t\mbox{-design}\},\\ 
s(C)&:=\max\{t\in \mathbb{N}\mid \exists w \mbox{ s.t.~} 
D_{w} \mbox{ is a } t\mbox{-design}\}.\end{aligned}$$ 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 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 IV code. If the case $\delta(C) < s(C)$ occurs, then $C$ is an extremal Type III $[764,382,192]$ code which has $\delta(C)=3$ and $s(C)=4,5$ or $6$.

This paper is organized as follows. In Section \[sec:pre\], we give definitions and some basic properties of self-dual codes and $t$-designs and review the concept of harmonic weight enumerators and some lemmas which are used in the proof of the main results. In Section \[sec: proof of thm1.3\] and \[sec: proof of thm1.4\], we give proofs of Theorem \[thm:main upper bound III\] and \[thm:main upper bound IV\], respectively.

All computer calculations in this paper were done with the help of Mathematica [@Mathematica].

Preliminaries {#sec:pre}
=============

Codes and the support $t$-designs
---------------------------------

Let ${{\mathbb F}}_q$ be the finite field of $q$ elements. A linear code $C$ of length $n$ is a linear subspace of ${{\mathbb F}}_{q}^{n}$. For $q=3$, an inner product $({x},{y})$ on ${{\mathbb F}}_q^n$ is given by $$(x,y)=\sum_{i=1}^nx_iy_i,$$ where $x,y\in {{\mathbb F}}_q^n$ with $x=(x_1,x_2,\ldots, x_n)$ and $y=(y_1,y_2,\ldots, y_n)$. The Hermitian inner product $({x},{y})$ on ${{\mathbb F}}_4^n$ is given by $$(x,y)_H=\sum_{i=1}^nx_iy_i^2,$$ where $x,y\in {{\mathbb F}}_4^n$ with $x=(x_1,x_2,\ldots, x_n)$ and $y=(y_1,y_2,\ldots, y_n)$. The dual of a linear code $C$ is defined as follows: for $q=3$, $$C^{\perp}=\{{y}\in {{\mathbb F}}_{q}^{n}\ | \ ({x},{y}) =0\ \mbox{ for all }{x}\in C\},$$ for $q=4$, $$C^{\perp,H}=\{{y}\in {{\mathbb F}}_{q}^{n}\ | \ ({x},{y})_H =0\ \mbox{ for all }{x}\in C\}.$$ A linear code $C$ is called self-dual if $C=C^{\perp}$ for $q=3$ and if $C=C^{\perp,H}$ for $q=4$. For $x \in{{\mathbb F}}_q^n$, the weight ${{\rm wt}}(x)$ is the number of its nonzero components. The minimum distance of a code $C$ is $\min\{{{\rm wt}}( x)\mid  x \in C,  x \neq  0 \}$. A linear code of length $n$, dimension $k$, and minimum distance $d$ is called an $[n,k,d]$ code. In this paper, we consider the following self-dual codes [@CS]:

Type III: A code is defined over ${{\mathbb F}}_{3}^{n}$ with all weights divisible by $3$,\
Type IV: A code is defined over ${{\mathbb F}}_{4}^{n}$ with all weights divisible by $2$.

A $t$-$(v,k,{\lambda})$ design (or $t$-design for short) is a pair $\mathcal{D}=(X,\mathcal{B})$, where $X$ is a set of points of cardinality $v$, and $\mathcal{B}$ a collection of $k$-element subsets of $X$ called blocks, with the property that any $t$ points are contained in precisely $\lambda$ blocks.

The support of a nonzero vector $ x:=(x_{1}, \dots, x_{n})$, $x_{i} \in {{\mathbb F}}_{q} = \{ 0,1, \dots, q-1 \}$ is the set of indices of its nonzero coordinates: ${\rm supp} ( x) = \{ i \mid x_{i} \neq 0 \}$. The support design of a code of length $n$ for a given nonzero weight $w$ is the design with $n$ points of coordinate indices, and blocks the supports of all codewords of weight $w$.

The following lemma is easily seen.

\[lem: divisible\] Let $\lambda(S)$ be the number of blocks containing a given set $S$ of $s$ points in a $t$-$(v,k,\lambda)$ design, where $0\leq s\leq t$. Then $$\lambda(S)\binom{k-s}{t-s}
=
\lambda\binom{v-s}{t-s}.$$ In particular, the number of blocks is $$\frac{v(v-1)\cdots(v-t+1)}{k(k-1)\cdots(k-t+1)}\lambda.$$

Harmonic weight enumerators {#sec:Har}
---------------------------

In this section, we extend a method of the harmonic weight enumerators which were used by Bachoc [@Bachoc] and Bannai et al. [@Bannai-Koike-Shinohara-Tagami]. For the readers convenience we quote from [@Bachoc; @Delsarte] the definitions and properties of discrete harmonic functions (for more information the reader is referred to [@Bachoc; @Delsarte]).

Let $\Omega=\{1, 2,\ldots,n\}$ be a finite set (which will be the set of coordinates of the code) and let $X$ be the set of its subsets, while, for all $k= 0,1, \ldots, n$, $X_{k}$ is the set of its $k$-subsets. We denote by ${{\mathbb R}}X$, ${{\mathbb R}}X_k$ the free real vector spaces spanned by respectively the elements of $X$, $X_{k}$. An element of ${{\mathbb R}}X_k$ is denoted by $$f=\sum_{z\in X_k}f(z)z$$ and is identified with the real-valued function on $X_{k}$ given by $z \mapsto f(z)$. Such an element $f\in {{\mathbb R}}X_k$ can be extended to an element $\widetilde{f}\in {{\mathbb R}}X$ by setting, for all $u \in X$, $$\widetilde{f}(u)=\sum_{z\in X_k, z\subset u}f(z).$$ If an element $g \in {{\mathbb R}}X$ is equal to some $\widetilde{f}$, for $f \in {{\mathbb R}}X_{k}$, we say that $g$ has degree $k$. The differentiation $\gamma$ is the operator defined by linearity from $$\gamma(z) =\sum_{y\in X_{k-1},y\subset z}y$$ for all $z\in X_k$ and for all $k=0,1, \ldots n$, and $\operatorname{Harm}_{k}$ is the kernel of $\gamma$: $$\operatorname{Harm}_k =\ker(\gamma|_{{{\mathbb R}}X_k}).$$

\[thm:design\] A set $\mathcal{B} \subset X_{m}$, where $m \leq n$, of blocks is a $t$-design if and only if $\sum_{b\in \mathcal{B}}\widetilde{f}(b)=0$ for all $f\in \operatorname{Harm}_k$, $1\leq k\leq t$.

In [@Bachoc], the harmonic weight enumerator associated to a linear code $C$ was defined as follows:

Let $C$ be a linear code of length $n$ and let $f\in\operatorname{Harm}_{k}$. The harmonic weight enumerator associated to $C$ and $f$ is $$W_{C,f}(x,y)=\sum_{c\in C}\widetilde{f}(c)x^{n-{{\rm wt}}(c)}y^{{{\rm wt}}(c)}.$$

Then the structure of these invariant rings is described as follows:

\[thm:invariant\]

1.  Let $C$ be a Type III code of length $n$, and let $f \in \operatorname{Harm}_{k}$. Then we have $W_{C,f}(x,y) =(xy)^{k} Z_{C,f} (x,y)$. Moreover, the polynomial $Z_{C,f} (x,y)$ is degree of $n-2k$ and is in $I_{G_3, \chi_{u,v}}$, where $u\equiv k\pmod{2}$ and $v\equiv -k\pmod{3}$, $$\begin{aligned}
    I_{G_3,\chi_{u,v}}=
    \left\{
    \begin{array}{ll}
    \langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(0,0),\\
    p_{4}\langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(0,1),\\
    p_{4}^2\langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(0,2),\\
    p_{6}\langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(1,0),\\
    p_{4}p_6\langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(1,1),\\
    p_{4}^2p_6\langle g_4,g_{12}\rangle &\mbox{ if }(u,v)=(1,2), 
    \end{array}
    \right. \end{aligned}$$ and $$\begin{aligned}
    \left\{
    \begin{array}{l}
    p_{4}=y(x^3-y^3), \\
    p_{6}=x^6-20x^3y^3-8y^6, \\
    g_{4}=x^4+8xy^3, \\
    g_{12}=y^3(x^3-y^3)^3. 
    \end{array} 
    \right. \end{aligned}$$

2.  Let $C$ be a Type IV code of length $n$, and let $f \in \operatorname{Harm}_{k}$. Then we have $W_{C,f}(x,y) =(xy)^{k} Z_{C,f} (x,y)$. Moreover, the polynomial $Z_{C,f} (x,y)$ is degree of $n-2k$ and is in $I_{G_4, \chi_{u,v}}$, where $u\equiv k\pmod{2}$ and $v\equiv k\pmod{2}$, $$\begin{aligned}
    I_{G_4,\chi_{u,v}}=
    \left\{
    \begin{array}{ll}
    \langle h_2,h_{6}\rangle &\mbox{ if }(u,v)=(0,0),\\
    q_{3}r_3\langle h_2,h_{6}\rangle &\mbox{ if }(u,v)=(1,1), 
    \end{array}
    \right. \end{aligned}$$ and $$\begin{aligned}
    \left\{
    \begin{array}{l}
    h_{2}=x^2+3y^2, \\
    h_{6}=y^2(x^2-y^2)^2, \\
    q_{3}=y(x^2-y^2), \\
    r_{3}=x^3-9xy^2. 
    \end{array}
    \right. \end{aligned}$$

We recall the slightly more general definition of the notion of a $T$-design, for a subset $T$ of $\{ 1,2, \ldots, n \}$: a set $\mathcal{B}$ of blocks is called a $T$-design if and only if $\sum_{b\in \mathcal{B}}\tilde{f}(b)=0$ for all $f\in \operatorname{Harm}_k$ and for all $k \in T$. By Theorem \[thm:design\], a $t$-design is a $T= \{1, \ldots, t \}$-design. Let $W_{C,f}=\sum_{i=0}^{n}c_f(i)x^{n-i}y^i$. Then we note that $D_w$ is a $T$-design if and only if $c_f(w)=0$ for all $f\in \mbox{Harm}_j$ with $j\in T$.

\[thm:[Calderbank-Delsarte]{}\]

1.  Let $D_{w}$ be the support design of weight $w$ of an extremal Type III code of length $n$ $($$n \geq 12$$)$.

    -   If $n \equiv 0 \pmod{12}$, $D_{w}$ is a $\{1, 2, 3, 4, 5, 7\}$-design.

    -   If $n \equiv 4 \pmod{12}$, $D_{w}$ is a $\{1, 2, 3, 5\}$-design.

    -   If $n \equiv 8 \pmod{12}$, $D_{w}$ is a $\{1, 3\}$-design.

2.  Let $D_{w}$ be the support design of weight $w$ of an extremal Type IV code of length $n$.

    -   If $n \equiv 0 \pmod{6}$ $($$n \geq 18$$)$, $D_{w}$ is a $\{1, 2, 3, 4, 5, 7\}$-design.

    -   If $n \equiv 2 \pmod{6}$, $D_{w}$ is a $\{1, 2, 3, 5\}$-design.

    -   If $n \equiv 4 \pmod{6}$, $D_{w}$ is a $\{1, 3\}$-design.

Coefficients of the harmonic weight enumerators of extremal Type III and Type IV codes
--------------------------------------------------------------------------------------

As we mentioned in Section \[sec:Har\], it is important for the support designs of a code $C$ whether the coefficients of $W_{C,f}(x,y)$ are zero or not. Therefore, we investigate it and show the following lemmas, where the binomial coefficient is defined by $$\binom{n}{k}=0$$ if $n<k$.

\[lem:poly. zero 1\]

1.  Let $Q_{1}=(x^{4}+8xy^{3})(x^{3}-y^{3})^{\alpha}$. If the coefficients of $x^{3\alpha+4-3i}y^{3i}$ in $Q_{1}$ are equal to $0$ for $0 \leq i \leq \alpha+1$, then $\alpha=9i-1$.

2.  Let $Q_{2}=(x^{6}-20x^{3}y^{3}-8y^{6})(x^{3}-y^{3})^{\alpha}$. Then the coefficients of $x^{3\alpha+6-3i}y^{3i}$ in $Q_{2}$ are not equal to $0$.

\(1) We have $$\begin{aligned}
Q_{1}&=(x^{4}+8xy^{3})(x^{3}-y^{3})^{\alpha} \\
     &=\sum_{i=0}^{\alpha+1} (-1)^{i} \left( \binom{\alpha}{i}-8\binom{\alpha}{i-1} \right)x^{3\alpha+4-3i}y^{3i}.\end{aligned}$$ If the coefficients of $x^{3\alpha+4-3i}y^{3i}$ in $Q_{1}$ are equal to $0$, $$\binom{\alpha}{i}-8\binom{\alpha}{i-1}=0.$$ Then we have $$\begin{aligned}
&\frac{\alpha !}{i !(\alpha-i)!}-8\frac{\alpha !}{(i-1) !(\alpha-i+1)!}=0 \\
\Leftrightarrow\ &\alpha-i+1-8i=0 \\
\Leftrightarrow\ &\alpha=9i-1.\end{aligned}$$

(2)We have $$\begin{aligned}
Q_{2}&=(x^{6}-20x^{3}y^{3}-8y^{6})(x^{3}-y^{3})^{\alpha} \\
     &=\sum_{i=0}^{\alpha+2} (-1)^{i} \left( \binom{\alpha}{i}+20\binom{\alpha}{i-1}-8\binom{\alpha}{i-2} \right)x^{3\alpha+6-3i}y^{3i}.\end{aligned}$$ If the coefficients of $x^{3\alpha+6-3i}y^{3i}$ in $Q_{2}$ are equal to $0$, $$\binom{\alpha}{i}+20\binom{\alpha}{i-1}-8\binom{\alpha}{i-2}=0.$$ Then we have $$\begin{aligned}
&\frac{\alpha !}{i !(\alpha-i)!}+20\frac{\alpha !}{(i-1) !(\alpha-i+1)!}-8\frac{\alpha !}{(i-2) !(\alpha-i+2)!}=0 \\
\Leftrightarrow\ &(\alpha-i+2)(\alpha-i+1)+20i(\alpha-i+2)-8i(i-1)=0 \\
\Leftrightarrow\ &\alpha^{2}+(18i+3)\alpha-27i^{2}+45i+2=0.\end{aligned}$$ We have $$\alpha=\frac{-18i-3 \pm \sqrt{432i^{2}-72i+1}}{2}.$$ Since $432i^{2}-72i+1$ is not a square number for $i>0$, $\alpha$ is not a positive integer, a contradiction. Thus the coefficients of $x^{3\alpha+6-3i}y^{3i}$ in $Q_{2}$ are not equal to $0$.

\[lem:poly. zero 2\]

1.  Let $R_{1}=(x^{2}+3y^{2})(x^{2}-y^{2})^{\alpha}$. If the coefficients of $x^{2\alpha+2-2i}y^{2i}$ in $R_{1}$ are equal to $0$ for $0 \leq i \leq \alpha+1$, then $\alpha=4i-1$.

2.  Let $R_{2}=(x^{3}-9xy^{2})(x^{2}-y^{2})^{\alpha}$. If the coefficients of $x^{2\alpha+3-2i}y^{2i}$ in $R_{2}$ are not equal to $0$.

3.  Let $R_{3}=(x^{2}+3y^{2})^{2}(x^{2}-y^{2})^{\alpha}$. If the coefficients of $x^{2\alpha+4-2i}y^{2i}$ in $R_{3}$ are equal to $0$ for $0 \leq i \leq \alpha+2$, then $48\alpha+112$ is a square number.

\(1) We have $$\begin{aligned}
R_{1}&=(x^{2}+3y^{2})(x^{2}-y^{2})^{\alpha} \\
 &=\sum_{i=0}^{\alpha+1} (-1)^{i} \left( \binom{\alpha}{i}-3\binom{\alpha}{i-1} \right)x^{2\alpha+2-2i}y^{2i}.\end{aligned}$$ If the coefficients of $x^{2\alpha+2-2i}y^{2i}$ in $R_{1}$ are equal to $0$, $$\binom{\alpha}{i}-3\binom{\alpha}{i-1}=0.$$ Then we have $$\begin{aligned}
&\frac{\alpha !}{i !(\alpha-i)!}-3\frac{\alpha !}{(i-1) !(\alpha-i+1)!}=0 \\
\Leftrightarrow\ &\alpha-i+1-3i=0 \\
\Leftrightarrow\ &\alpha=4i-1.\end{aligned}$$

\(2) We have $$\begin{aligned}
R_{2}&=(x^{3}-9xy^{2})(x^{2}-y^{2})^{\alpha} \\
 &=\sum_{i=0}^{\alpha+1} (-1)^{i} \left( \binom{\alpha}{i}+9\binom{\alpha}{i-1} \right)x^{2\alpha+3-2i}y^{2i}.\end{aligned}$$ If the coefficients of $x^{2\alpha+3-2i}y^{2i}$ in $R_{2}$ are equal to $0$, $$\binom{\alpha}{i}+9\binom{\alpha}{i-1}=0.$$ Then we have $$\begin{aligned}
&\frac{\alpha !}{i !(\alpha-i)!}+9\frac{\alpha !}{(i-1) !(\alpha-i+1)!}=0 \\
\Leftrightarrow\ &\alpha-i+1+9i=0 \\
\Leftrightarrow\ &\alpha=-8i-1<0.\end{aligned}$$ Thus the coefficients of $x^{2\alpha+3-2i}y^{2i}$ in $R_{2}$ are not equal to $0$.

\(3) We have $$\begin{aligned}
R_{3}&=(x^{2}+3y^{2})^{2}(x^{2}-y^{2})^{\alpha} \\
 &=\sum_{i=0}^{\alpha+1} (-1)^{i} \left( \binom{\alpha}{i}-6\binom{\alpha}{i-1}+9\binom{\alpha}{i-2} \right)x^{2\alpha+4-2i}y^{2i}.\end{aligned}$$ If the coefficients of $x^{2\alpha+4-2i}y^{2i}$ in $R_{3}$ are equal to $0$, $$\binom{\alpha}{i}-6\binom{\alpha}{i-1}+9\binom{\alpha}{i-2}=0.$$ Then we have $$\begin{aligned}
&\frac{\alpha !}{i !(\alpha-i)!}-6\frac{\alpha !}{(i-1) !(\alpha-i+1)!}+9\frac{\alpha !}{(i-2) !(\alpha-i+2)!}=0 \\
\Leftrightarrow\ &(\alpha-i+2)(\alpha-i+1)-6i(\alpha-i+2)+9i(i-1)=0 \\
\Leftrightarrow\ &16i^{2}-(8\alpha+24)i+\alpha^{2}+3\alpha+2=0.\end{aligned}$$ We have $$i=\frac{4\alpha+12 \pm \sqrt{48\alpha+112}}{16}.$$ Since $i$ is an integer, $48\alpha+112$ is a square number.

Proof of Theorem \[thm:main upper bound III\] {#sec: proof of thm1.3}
=============================================

Case for $n=12m$ {#sec: case for n=12m}
----------------

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 $t$-design of weight $w$ of an extremal Type III code of length $n=12m$. Then all $D_{w}^{12m}$ are $8$-designs simultaneously, or none of $D_{w}^{12m}$ is an $8$-design.

Let us assume that $t=8$, and $C$ is an extremal Type III $[12m,6m,3m+3]$ code. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) (xy)^{8} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{3},\chi_{0,1}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = p_{4} \sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-5} g_{12}^{i}.$$ Since the minimum weight of $C$ is $3m+3$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{8} p_{4}g_{4} g_{12}^{m-2}   \\
& =c(f) (xy)^{8} y^{3m-5} (x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-5}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (1), the coefficients of $x^{9m-11-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-5}$ are not equal to $0$ for $0 \leq i \leq 3m-4$ since $3m-5 \neq 9i-1$. Therefore, all $D_{w}^{12m}$ are $8$-designs simultaneously, or none of $D_{w}^{12m}$ is an $8$-design.

By Proposition \[prop:type3 12m not 8-design\] and \[prop:type3 12m harm\], we obtain the following theorem.

\[thm:main thm 12m\]

If $D_{w}^{12m}$ becomes a $7$-design for any $w$, then $m$ must be in the set $\{15,38,43,64 \}$.

$D_{w}^{12m}$ is never an $8$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound III\] (1) is completed.

Case for $12m+4$ {#sec: case for n=12m+4}
----------------

In this section, we consider the case of extremal Type III $[12m+4,6m+2,3m+3]$ codes ($m \leq 74$). Let $C$ be an extremal Type III $[12m+4,6m+2,3m+3]$ code, and $D_{3m+3}^{12m+4}$ 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 $$2(12m+4)(12m+3)(12m+2)\frac{(4m)!}{m!(3m+3)!}.$$ Therefore, by the Assmus–Mattson theorem, $D_{3m+3}^{12m+4}$ is a $3$-design with parameters $$\left(12m+4,3m+3,\binom{4m}{m} \right).$$

\[prop:type3 12m+4 not 7-design\] Let $D_{3m+3}^{12m+4}$ be the support $t$-design of the minimum weight of an extremal Type III code of length $n=12m+4$.

1.  If $t \geq 4$, then $D_{3m+3}^{12m+4}$ is a $5$-design and $m$ must be in the set\
    $\{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}$.

2.  If $D_{3m+3}^{12m+4}$ is a $6$-design, $m$ must be $67$.

3.  $D_{3m+3}^{12m+4}$ is never a $7$-design.

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (1), $D_{3m+3}^{12m+4}$ is a $5$-design if $t \geq 4$. If $D_{3m+3}^{12m+4}$ is a $5$-design, by Lemma \[lem: divisible\], $$\lambda_{4}=\frac{3m}{12m+1}\binom{4m}{m} \mbox{ and } 
\lambda_{5}=\frac{3m(3m-1)}{(12m+1)12m}\binom{4m}{m}$$ are positive integers. By a computation for $m \leq 74$, if $\lambda_{4}$ and $\lambda_{5}$ are positive integers, then we have $$m \in \{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}.$$

(2)If $D_{3m+3}^{12m+4}$ is a $6$-design, by Lemma \[lem: divisible\], $$\lambda_{6}=\frac{3m(3m-1)(3m-2)}{(12m+1)12m(12m-1)}\binom{4m}{m}$$ is a positive integer. Then we have $m=67$.

\(3) We have checked that $$\lambda_{7}=\frac{3m(3m-1)(3m-2)(3m-3)}{(12m+1)12m(12m-1)(12m-2)}\binom{4m}{m}$$ is not a positive integer for $m=67$. Therefore, by Lemma \[lem: divisible\], $D_{3m+3}^{12m+4}$ is never a $7$-design.

For $t \geq 6$, we give the following proposition.

\[prop:type3 12m+4 harm\] Let $D_{w}^{12m+4}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+4$.

1.  All $D_{w}^{12m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $6$-design.

2.  All $D_{w}^{12m+4}$ are $7$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $7$-design.

Let $C$ be an extremal Type III $[12m+4,6m+2,3m+3]$ code.

\(1) Let us assume that $t=6$. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) (xy)^{6} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{3},\chi_{0,0}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) =  \sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-2} g_{12}^{i}.$$ Since the minimum weight of $C$ is $3m+3$, we have $a_{i}=0$ for $i \neq m-1$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{6} g_{4} g_{12}^{m-1}  \\
& =c(f) (xy)^{6} y^{3m-3} (x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-3}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (1), the coefficients of $x^{9m-5-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-3}$ are not equal to $0$ for $0 \leq i \leq 3m-2$ since $3m-3 \neq 9i-1$. Therefore, all $D_{w}^{12m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $6$-design.

\(2) Let us assume that $t=7$. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) (xy)^{7} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{3},\chi_{1,2}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = p_{4}^{2}p_{6} \sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-6} g_{12}^{i}.$$ Since the minimum weight of $C$ is $3m+3$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{7}p_{4}^{2}p_{6} g_{12}^{m-2}  \\
& =c(f) (xy)^{7} y^{3m-4} (x^{6}-20x^{3}y^{3}-8y^{6}) (x^{3}-y^{3})^{3m-4}. \end{aligned}$$

By Lemma \[lem:poly. zero 1\] (2), the coefficients of $x^{9m-6-3i}y^{3i}$ in $(x^{6}-20x^{3}y^{3}-8y^{6}) (x^{3}-y^{3})^{3m-4}$ are not equal to $0$ for $0 \leq i \leq 3m-2$. Therefore, all $D_{w}^{12m+4}$ are $7$-designs simultaneously, or none of $D_{w}^{12m+4}$ is a $7$-design.

By Proposition \[prop:type3 12m+4 not 7-design\] and \[prop:type3 12m+4 harm\], we obtain the following theorem.

\[thm:main thm 12m+4\] Let $D_{w}^{12m+4}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+4$ $($$m \leq 74$$)$.

1.  If $D_{w}^{12m+4}$ becomes a $5$-design for any $w$, then $m$ must be in the set\
    $\{11,18,21,25,32,39,43,46,49,54,60,65,67,68,74 \}$.

2.  If $D_{w}^{12m+4}$ becomes a $6$-design for any $w$, then $m$ must be $67$.

3.  In the case $m=67$, $D_{w}^{808}$ is a $3,5$ or $6$-design for any $w$.

4.  $D_{w}^{12m+4}$ is never a $7$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound III\] (2) is completed.

Case for $12m+8$ {#sec: case for n=12m+8}
----------------

In this section, we consider the case of extremal Type III $[12m+8,6m+4,3m+3]$ codes ($m \leq 77$). Let $C$ be an extremal Type III $[12m+8,6m+4,3m+3]$ code, and $D_{3m+3}^{12m+8}$ 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 $$6(12m+8)\frac{(4m+2)!}{m!(3m+3)!}.$$ Therefore, by the Assmus–Mattson theorem, $D_{3m+3}^{12m+8}$ is a $1$-design with parameters $$\left(12m+8,3m+3,3\binom{4m+2}{m} \right).$$

\[prop:type3 12m+8 not 4-design\] Let $D_{3m+3}^{12m+8}$ be the support $t$-design of the minimum weight of an extremal Type III code of length $n=12m+8$.

1.  If $t \geq 2$, then $D_{3m+3}^{12m+8}$ is a $3$-design and $m$ must be in the set $\{ 14,37,42,63 \}$.

2.  $D_{3m+3}^{12m+8}$ is never a $4$-design.

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (1), $D_{3m+3}^{12m+8}$ is a $3$-design if $t \geq 2$. If $D_{3m+3}^{12m+8}$ is a $3$-design, by Lemma \[lem: divisible\], $$\lambda_{2}=\frac{3m+2}{12m+7}3\binom{4m+2}{m}\ {\rm and}\ 
\lambda_{3}=\frac{(3m+2)(3m+1)}{(12m+7)(12m+6)}3\binom{4m+2}{m}$$ are positive integers. By a computation for $m \leq 77$, if $\lambda_{2}$ and $\lambda_{3}$ are positive integers, then we have $m \in \{ 14,37,42,63 \}$.

\(2) We have checked that $$\lambda_{4}=\frac{(3m+2)(3m+1)3m}{(12m+7)(12m+6)(12m+5)}3\binom{4m+2}{m}$$ is not a positive integer for $m \in \{ 14,37,42,63 \}$. Therefore, by Lemma \[lem: divisible\], $D_{3m+3}^{12m+8}$ is never a $4$-design.

We have the following proposition.

\[prop:type3 12m+8 harm\] Let $D_{w}^{12m+8}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+8$.

1.  If $m \not\equiv 0 \pmod 3$, all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design.

2.  Assume that $m \equiv 0 \pmod 3$. If $w \neq 4m+3$, then all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design. In the case $w=4m+3$, $D_{4m+3}^{12m+8}$ is a $\{1,3,4 \}$-design.

3.  In the case $m=63$ and $w=255$, $D_{255}^{764}$ is not a $7$-design.

Let $C$ be an extremal Type III $[12m+8,6m+4,3m+3]$ code. Let us assume that $t=4$. Then by the Theorem \[thm:invariant\] (1) we have $W_{C,f}(x,y) =c(f) (xy)^{4} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{3},\chi_{0,2}}$. By Theorem \[thm:invariant\] (1), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) =  p_{4}^{2}\sum_{i=0}^{m}a_{i}g_{4}^{3(m-i)-2} g_{12}^{i}.$$ Since the minimum weight of $C$ is $3m+3$, we have $a_{i}=0$ for $i \neq m-1$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{4} p_{4}^{2}g_{4} g_{12}^{m-1}  \\
& =c(f) (xy)^{4} y^{3m-1} (x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-1}. \end{aligned}$$

\(1) By Lemma \[lem:poly. zero 1\] (1), if $m \not\equiv 0 \pmod 3$, the coefficients of $x^{9m+1-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-1}$ are not equal to $0$ for $0 \leq i \leq 3m$ since $3m-3 \neq 9i-1$. Therefore, all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design for $m \not\equiv 0 \pmod 3$.

\(2) Assume that $m \equiv 0 \pmod 3$. By Lemma \[lem:poly. zero 1\] (1), if the coefficients of $x^{9m+1-3i}y^{3i}$ in $(x^{4}+8xy^{3}) (x^{3}-y^{3})^{3m-1}$ are equal to $0$ for $0 \leq i \leq 3m$, then $m=3i$. Hence the coefficients of $x^{8m+5}y^{4m+3}$ in $W_{C,f}(x,y) $ are equal to $0$. Therefore, $D_{4m+3}^{12m+8}$ is a $\{1,3,4 \}$-design. If $w \neq 4m+3$, then all $D_{w}^{12m+8}$ are $4$-designs simultaneously, or none of $D_{w}^{12m+8}$ is a $4$-design.

\(3) Let $C'$ be an extremal Type III $[764,382,192]$ code and $C'_{255}$ be the set of codewords of weight 255 of $C'$. By [@mallows-sloane], we have $$\begin{aligned}
|C'_{255}|=& 57722041604247479907056082274041845325097239194558706847\\
& 5581740325339465514100889807420805771366809484288.\end{aligned}$$ Let $D_{255}^{764}$ be a $t$-$(764,255,\lambda_{t})$ design. Then we have $$\lambda_{t}=\frac{255\cdot254\cdots(255-t+1)}{764\cdot763\cdots(764-t+1)}|C'_{255}|.$$ We have checked that $\lambda_{t}$ is a positive integer for $t=1,2,\ldots, 6$ and $\lambda_{7}$ is not a positive integer. Thus $D_{255}^{764}$ is not a $7$-design.

By Proposition \[prop:type3 12m+8 not 4-design\] and \[prop:type3 12m+8 harm\], we obtain the following theorem.

\[thm:main thm 12m+8\] Let $D_{w}^{12m+8}$ be the support $t$-design of weight $w$ of an extremal Type III code of length $n=12m+8$ $($$m \leq 77$$)$.

1.  If $D_{w}^{12m+8}$ becomes a $3$-design for any $w$, then $m$ must be in the set $\{ 14,37,42,63 \}$.

2.  In the case $m \in \{ 14,37,42 \}$, $D_{w}^{12m+8}$ is a $1$ or $3$-design for any $w$.

3.  Assume that $m=63$. If $w \neq 255$, $D_{w}^{764}$ is a $1$ or $3$-design and $D_{255}^{764}$ is a $1, 4, 5$ or $6$-design.

4.  $D_{w}^{12m+8}$ is never a $7$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound III\] (3) is completed.

Proof of Theorem \[thm:main upper bound IV\] {#sec: proof of thm1.4}
============================================

Case for $n=6m$ {#sec: case for n=6m}
---------------

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 $D_{w}^{6m}$ are $8$-designs simultaneously, or none of $D_{w}^{6m}$ is an $8$-design.

Let $C$ be an extremal Type IV $[6m,3m,2m+2]$ code. Let us assume that $t=8$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{8} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-8} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-3$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{8} h_{2} h_{6}^{m-3}   \\
& =c(f) (xy)^{8} y^{2m-6} (x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-6}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (1), the coefficients of $x^{4m-10-2i}y^{2i}$ in $(x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-6}$ are not equal to $0$ for $0 \leq i \leq 2m-5$ since $2m-6 \neq 4i-1$. Therefore, all $D_{w}^{6m}$ are $8$-designs simultaneously, or none of $D_{w}^{6m}$ is an $8$-design.

By Proposition \[prop:type4 6m not 8-design\] and \[prop:type4 6m harm\], we obtain the following theorem.

\[thm:main thm 6m\]

If $D_{w}^{6m}$ becomes a $7$-design for any $w$, then $m$ must be in the set $\{10,15 \}$.

$D_{w}^{6m}$ is never an $8$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound IV\] (1) is completed.

Case for $n=6m+2$ {#sec: case for n=6m+2}
-----------------

In this section, we consider the case of extremal Type IV $[6m+2,3m+1,2m+2]$ codes ($m \leq 19$). Let $C$ be an extremal Type IV $[6m+2,3m+1,2m+2]$ code, and $D_{2m+2}^{6m+2}$ be the support (with duplicates omitted) design of the minimum weight of $C$. By [@MOSW1978 Theorem 14], the number of codewords of minimum nonzero weight of $C$ is equal to $$\frac{3(6m+1)}{m+1}\binom{3m+1}{m}.$$ Therefore, by the Assmus–Mattson theorem, $D_{2m+2}^{6m+2}$ is a $3$-design with parameters $$\left(6m+2,2m+2,\frac{1}{3}\binom{3m}{m} \right).$$

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

1.  If $t \geq 4$, then $D_{2m+2}^{6m+2}$ is a $5$-design and $m$ must be $11$.

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

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (2), $D_{2m+2}^{6m+2}$ is a $5$-design if $t \geq 4$. If $D_{2m+2}^{6m+2}$ is a $5$-design, by Lemma \[lem: divisible\], $$\lambda_{4}=\frac{2m-1}{6m-1}\frac{1}{3}\binom{3m}{m} \mbox{ and } 
\lambda_{5}=\frac{(2m-1)(2m-2)}{(6m-1)(6m-2)}\frac{1}{3}\binom{3m}{m}$$ are positive integers. By a computation for $m \leq 19$, if $\lambda_{4}$ and $\lambda_{5}$ are positive integers, then we have $m=11$.

\(2) For $m=11$, we have checked that $$\begin{aligned}
\lambda_{6}&=\frac{(2m-1)(2m-2)(2m-3)}{(6m-1)(6m-2)(6m-3)}\frac{1}{3}\binom{3m}{m},\mbox{ and }\\
\lambda_{7}&=\frac{(2m-1)(2m-2)(2m-3)(2m-4)}{(6m-1)(6m-2)(6m-3)(6m-4)}\frac{1}{3}\binom{3m}{m}\end{aligned}$$ are positive integers and $$\lambda_{8}=\frac{(2m-1)(2m-2)(2m-3)(2m-4)(2m-5)}{(6m-1)(6m-2)(6m-3)(6m-4)(6m-5)}\frac{1}{3}\binom{3m}{m}$$ is not a positive integer. Therefore, by Lemma \[lem: divisible\], $D_{2m+2}^{6m+2}$ is never an $8$-design.

For $t \geq 6$, we give the following proposition.

\[prop:type4 6m+2 harm\] Let $D_{w}^{6m+2}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m+2$.

1.  All $D_{w}^{6m+2}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $6$-design.

2.  All $D_{w}^{6m+2}$ are $7$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $7$-design.

3.  In the case $m=11$. All $D_{w}^{68}$ are $8$-designs simultaneously, or none of $D_{w}^{68}$ is an $8$-design.

Let $C$ be an extremal Type IV $[6m+2,3m+1,2m+2]$ code.

\(1) Let us assume that $t=6$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{6} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-5} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{6} h_{2} h_{6}^{m-2}   \\
& =c(f) (xy)^{6} y^{2m-4} (x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-4}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (1), the coefficients of $x^{4m-6-2i}y^{2i}$ in $(x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-4}$ are not equal to $0$ for $0 \leq i \leq 2m-3$ since $2m-4 \neq 4i-1$. Therefore, all $D_{w}^{6m+2}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $6$-design.

\(2) Let us assume that $t=7$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{7} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{1,1}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = q_{3}r_{3}\sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-9} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-3$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{7} q_{3}r_{3} h_{6}^{m-3}   \\
& =c(f) (xy)^{7} y^{2m-5} (x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-5}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (2), the coefficients of $x^{4m-7-2i}y^{2i}$ in $(x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-5}$ are not equal to $0$ for $0 \leq i \leq 2m-4$. Therefore, all $D_{w}^{6m+2}$ are $7$-designs simultaneously, or none of $D_{w}^{6m+2}$ is a $7$-design.

\(3) Let us assume that $t=8$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{8} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-7} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-3$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{8} h_{2}^{2} h_{6}^{m-3}   \\
& =c(f) (xy)^{8} y^{2m-6} (x^{2}+3y^{2})^{2} (x^{2}-y^{2})^{2m-6}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (3), if the coefficients of $x^{4m-8-2i}y^{2i}$ in $(x^{2}+3y^{2})^{2} (x^{2}-y^{2})^{2m-6}$ are equal to $0$, then $48(2m-6)+112$ is a square number. In the case $m=11$, $48(2m-6)+112=880$ is not a square number. Therefore, all $D_{w}^{68}$ are $8$-designs simultaneously, or none of $D_{w}^{68}$ is an $8$-design.

By Proposition \[prop:type4 6m+2 not 8-design\] and \[prop:type4 6m+2 harm\], we obtain the following theorem.

\[thm:main thm 6m+2\] Let $D_{w}^{6m+2}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m+2$ $($$m \leq 19$$)$.

1.  If $D_{w}^{6m+2}$ becomes a $5$-design for any $w$, then $m$ must be $11$.

2.  In the case $m=11$, $D_{w}^{68}$ is a $3,5,6$ or $7$-design for any $w$.

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

Thus the proof of Theorem \[thm:main upper bound IV\] (2) is completed.

Case for $n=6m+4$ {#sec: case for n=6m+4}
-----------------

In this section, we consider the case of extremal Type IV $[6m+4,3m+2,2m+2]$ codes ($m \leq 21$). Let $C$ be an extremal Type IV $[6m+4,3m+2,2m+2]$ code, and $D_{2m+2}^{6m+4}$ be the support (with duplicates omitted) design of the minimum weight of $C$. By [@MOSW1978 Theorem 14], the number of codewords of minimum nonzero weight of $C$ is equal to $$3\binom{3m+2}{m+1}.$$ Therefore, by the Assmus–Mattson theorem, $D_{2m+2}^{6m+4}$ is a $1$-design with parameters $$\left(6m+4,2m+2,\binom{3m+1}{m} \right).$$

\[prop:type4 6m+4 not 6-design\] Let $D_{2m+2}^{6m+4}$ be the support $t$-design of the minimum weight of an extremal Type IV code of length $n=6m+4$.

1.  \[[(1)]{}\] If $D_{2m+2}^{6m+4}$ is a $3$-design, then $m$ must be in the set\
    $\{3,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21\}$.

2.  If $D_{2m+2}^{6m+4}$ is a $4$-design, then $m$ must be in the set $\{ 9,14,19 \}$.

3.  \[[(3)]{}\] If $D_{2m+2}^{6m+4}$ is a $5$-design, then $m$ must be in the set $\{ 14,19 \}$.

4.  \[[(4)]{}\] $D_{2m+2}^{6m+4}$ is never a $6$-design.

\(1) By Theorem \[thm:[Calderbank-Delsarte]{}\] (2), $D_{2m+2}^{6m+4}$ is a $3$-design if $t \geq 2$. If $m \in \{1,2,4,13\}$, then $D_{2m+2}^{6m+4}$ is not a $3$-design.

For $m\in \{1,4,13\}$. we have checked that both $$\lambda_{2}=\frac{2m+1}{6m+3}\binom{3m+1}{m} \mbox{ or } 
\lambda_{3}=\frac{(2m+1)2m}{(6m+3)(6m+2)}\binom{3m+1}{m}$$ are not positive integers.

For $m=2$, it is known that there is no $3$-$(16,6,2)$ design by [@Brouwer1977].

\(2) If $D_{2m+2}^{6m+4}$ is a $4$-design, $$\lambda_{4}=\frac{(2m+1)2m(2m-1)}{(6m+3)(6m+2)(6m+1)}\binom{3m+1}{m}$$ is a positive integer. By a computation for $m \leq 21$, if $\lambda_{4}$ is a positive integer, then we have $m \in \{ 9,14,19 \}$.

\(3) For $m \in \{ 9,14,19 \}$, if $$\lambda_{5}=\frac{(2m+1)2m(2m-1)(2m-2)}{(6m+3)(6m+2)(6m+1)6m}\binom{3m+1}{m}$$ is a positive integer, then $m \in \{ 14,19 \}$.

\(4) We have checked that $$\lambda_{6}=\frac{(2m+1)2m(2m-1)(2m-2)(2m-3)}{(6m+3)(6m+2)(6m+1)6m(6m-1)}\binom{3m+1}{m}$$ is not a positive integer for $m \in \{ 14,19 \}$.

For $t \geq 4$, we give the following proposition.

\[prop:type4 6m+4 harm\] Let $D_{w}^{6m+4}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m+4$.

1.  All $D_{w}^{6m+4}$ are $4$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $4$-design.

2.  All $D_{w}^{6m+4}$ are $5$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $5$-design.

3.  If $m\in \{ 14,19 \}$, all $D_{w}^{6m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $6$-design.

Let $C$ be an extremal Type IV $[6m+4,3m+2,2m+2]$ code.

\(1) Let us assume that $t=4$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{4} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-2} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-1$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{4} h_{2} h_{6}^{m-1}   \\
& =c(f) (xy)^{4} y^{2m-2} (x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-2}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (1), the coefficients of $x^{4m-2-2i}y^{2i}$ in $(x^{2}+3y^{2}) (x^{2}-y^{2})^{2m-2}$ are not equal to $0$ for $0 \leq i \leq 2m-1$ since $2m-2 \neq 4i-1$. Therefore, all $D_{w}^{6m+4}$ are $4$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $4$-design.

\(2) Let us assume that $t=5$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{5} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{1,1}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = q_{3}r_{3}\sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-6} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{5} q_{3}r_{3} h_{6}^{m-2}   \\
& =c(f) (xy)^{5} y^{2m-3} (x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-3}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (2), the coefficients of $x^{4m-3-2i}y^{2i}$ in $(x^{3}-9xy^{2}) (x^{2}-y^{2})^{2m-3}$ are not equal to $0$ for $0 \leq i \leq 2m-2$. Therefore, all $D_{w}^{6m+4}$ are $5$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $5$-design.

\(3) Let us assume that $t=6$. Then by the Theorem \[thm:invariant\] (2) we have $W_{C,f}(x,y) =c(f) (xy)^{6} Z_{C,f} (x,y)$, where $c(f)$ is a linear function from Harm$_{t}$ to ${{\mathbb R}}$ and $Z_{C,f} (x,y) \in I_{G_{4},\chi_{0,0}}$. By Theorem \[thm:invariant\] (2), $Z_{C,f} (x,y)$ can be written in the following form: $$Z_{C,f}(x,y) = \sum_{i=0}^{m}a_{i}h_{2}^{3(m-i)-4} h_{6}^{i}.$$ Since the minimum weight of $C$ is $2m+2$, we have $a_{i}=0$ for $i \neq m-2$. Therefore, $W_{C,f}(x,y)$ can be written in the following form: $$\begin{aligned}
W_{C,f}(x,y) &=c(f) (xy)^{6} h_{2}^{2} h_{6}^{m-2}   \\
& =c(f) (xy)^{6} y^{2m-4} (x^{2}+3y^{2})^{2} (x^{2}-y^{2})^{2m-4}. \end{aligned}$$

By Lemma \[lem:poly. zero 2\] (3), if the coefficients of $x^{4m-4-2i}y^{2i}$ in $(x^{2}+3y^{2})^{2} (x^{2}-y^{2})^{2m-4}$ are equal to $0$, then $48(2m-4)+112$ is a square number. If $m=14$ or $19$, $48(2m-4)+112=1264$ or $48(2m-4)+112=1744$ is not a square number. Therefore, if $m\in \{ 14,19 \}$, all $D_{w}^{6m+4}$ are $6$-designs simultaneously, or none of $D_{w}^{6m+4}$ is a $6$-design.

By Proposition \[prop:type4 6m+4 not 6-design\] and \[prop:type4 6m+4 harm\], we obtain the following theorem.

\[thm:main thm 6m+4\] Let $D_{w}^{6m+4}$ be the support $t$-design of weight $w$ of an extremal Type IV code of length $n=6m+4$ $($$m \leq 21$$)$.

1.  If $D_{w}^{6m+4}$ becomes a $3$-design for any $w$, then $m$ must be in the set\
    $\{3,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21\}$.

2.  If $D_{w}^{6m+4}$ becomes a $4$-design for any $w$, then $m$ must be in the set $\{ 9,14,19 \}$.

3.  In the case $m=9$, $D_{w}^{58}$ is a $1,3$ or $4$-design for any $w$. If $m \in \{ 14,19 \}$, $D_{w}^{6m+4}$ is a $1,3,4$ or $5$-design for any $w$.

4.  $D_{w}^{6m+4}$ is never a $6$-design for any $w$.

Thus the proof of Theorem \[thm:main upper bound IV\] (3) is completed.

Acknowledgments {#acknowledgments .unnumbered}
===============

The authors are supported by JSPS KAKENHI (18K03217).

[999]{}

E.F. Assmus, Jr. and H.F. Mattson, Jr., New $5$-designs, [*J. Combin. Theory*]{} [**6**]{} (1969), 122–151.

C. Bachoc, On harmonic weight enumerators of binary codes, [*Des. Codes Cryptogr.*]{} [**18**]{} (1999), no. 1-3, 11–28.

C. Bachoc, Harmonic weight enumerators of nonbinary codes and MacWilliams identities, [*Codes and association schemes*]{} (Piscataway, NJ, 1999), 1–23, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 56, Amer. Math. Soc., Providence, RI, 2001.

E. Bannai, M. Koike, M. Shinohara and M. Tagami, Spherical designs attached to extremal lattices and the modulo $p$ property of Fourier coefficients of extremal modular forms, [*Mosc. Math. J.*]{} [**6**]{} (2006), 225–264.

A.E. Brouwer, The $t$-designs with $v < 18$, Math. Centr. report ZN76, Amsterdam (Aug. 1977).

A.R. Calderbank, and P. Delsarte, On error-correcting codes and invariant linear forms, *SIAM J. Disc. Math.* [**6**]{} (1993), No.1, 1–23.

P.J. Cameron, J.H. van Lint, [*Designs, graphs, codes and their links*]{}, London Mathematical Society Student Texts, 22. Cambridge University Press, Cambridge, 1991.

J.H. Conway, N.J.A. Sloane, [*Sphere Packings Lattices and Groups*]{}, third edition, Springer, New York, 1999.

P. Delsarte, [Hahn polynomials, discrete harmonics, and $t$-designs]{}, [*SIAM J. Appl. Math.*]{} [**34**]{} (1978), no. 1, 157–166.

N. Horiguchi, T. Miezaki and H. Nakasora, On the support designs of extremal binary doubly even self-dual codes, *Des. Codes Cryptogr.*, [**72**]{} (2014), 529–537.

F.J. MacWilliams, A.M. Odlyzko, N.J.A. Sloane and H.N. Ward, Self-dual codes over $GF(4)$, *J. Combin. Theory Ser. A* [**25**]{} (1978), 288–318

C.L. Mallows and N.J.A. Sloane, An upper bound for self-dual codes, *Inform. Control* [**22**]{} (1973), 188–200.

T. Miezaki, A. Munemasa and H. Nakasora, A note on Assmus–Mattson type theorems, in preparation.

T. Miezaki and H. Nakasora, An upper bound of the value of $t$ of the support $t$-designs of extremal binary doubly even self-dual codes, *Des. Codes Cryptogr.*, [**79**]{} (2016), 37–46.

T. Miezaki and H. Nakasora, The support designs of the triply even binary codes of length $48$, *J. Combin. Designs*, [**27**]{} (2019), 673–681.

Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).

S. Zhang, On the nonexistence of extremal self-dual codes, [*Discrete Appl. Math.*]{} [**91**]{} (1999), 277–286.

[^1]: Faculty of Education, University of the Ryukyus, Okinawa 903–0213, Japan, E-mail: miezaki@edu.u-ryukyu.ac.jp (Corresponding author)

[^2]: Institute for Promotion of Higher Education, Kobe Gakuin University, Kobe 651–2180, Japan, E-mail: nakasora@ge.kobegakuin.ac.jp
