$C$ 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 