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
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As we mentioned in Section \[sec:Har\], it is important for the support designs of 