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 