 to the degree of $h(x)$.* $\Box$

Let $h_1(x)$ and $h_2(x)$ be the parity-check polynomials of $1$-generator skew GQC codes $C_1$ and $C_2$, respectively. If $C_1=C_2$, then $h_1(x)=h_2(x)$, which implies that ${\rm deg }(h_1(x))={\rm deg}(h_2(x))$. It means that $R/(h_1(x))_r=R/(h_2(x))_r$. Conversely, suppose $h_1(x)$ and $h_2(x)$ are similar. Then we have $R/(h_1(x))_r\cong R/(h_2(x))_r$, which implies that $C_1= C_2$. Then from the discussion above, we have $C_1=C_2$ if and only if $h_1(x)\sim h_2(x)$, i.e., any $1$-generator skew GQC code has a unique parity-check polynomial up to similarity.

[**Theorem 4.3**]{} *Let $C$ be a $1$-generator skew GQC code of block length $(m_1, m_2,\ldots, m_l)$ and length $\sum_{i=1}^lm_i$ generated by $c(x)=(c_1(x), c_2(x), \ldots,c_l(x))\in {\mathcal R}$. Suppose $h_i(x)$ is given as in Theorem 4.2 and $h(x)={\rm lclm}\{h_1(x),h_2(x),\ldots,h_l(x)\}$. Let $\delta_i$ denote the number of consecutive powers of a primitive $m_i$-th root of unity that among the right zeros of $(x^{m_i}-1)/h_i(x)$. Then\
(i) $d_{\rm H}(C)\geq \sum_{{i}\not \in K}(\delta_i+1)$, where $K\subseteq \{1,2,\ldots,l\}$ is a set of maximum size such that ${\rm lclm}_{i\in K}h_i(x)\neq h(x)$.\
(ii) If $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $d_{\rm H}(C)\geq\sum_{i=1}^l(\delta_i+1)$.*

*Proof* Let $a(x)\in C$ be a nonzero codeword. Then there exists a polynomial $f(x)\in R$ such that $a(x)=f(x)c(x)$. Since for each $i=1,2,\ldots,l$, the $i$-th component is zero if and only if $(x^{m_i}-1)\mid f(x)c_i(x)$, i.e., if and only if $h_i(x)\mid f(x)$. Therefore $a(x)=0$ if and only if $h(x)\mid f(x)$. So $a(x)\neq 0$ if and only if $h(x)\nmid f(x)$. This implies that $c(x)\neq 0$ has the most number of zero blocks whenever $h(x)\neq {\rm lclm}_{i\in K}h_i(x)$, where ${\rm lclm}_{i\in K}h_i(x)\mid f(x)$, and $K$ is a maximal subset of $\{1,2,\ldots,l\}$ having this property. Thus, $d_{\rm H}(C)\geq \sum_{i\notin K}d_i$, where $d_i=d_{\rm H}(\varphi_i(C))\geq \delta_i+1$. Clearly, $K=\emptyset$ if and only if $h