# Null Space of Parity Check Matrix

We know that if $\alpha$ be s a primitive element of $F_q$ where $q$ is a prime power then the null space of the following matrix generates a cyclic code of designed distance $\mu$[1]. $$G_{\alpha}^{(q,\mu)}=\left( \begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \alpha^1 & \alpha^2 & \cdots & \alpha^{q-2} & 0 \\ 1 & \alpha^2 & \alpha^4 & \cdots & \alpha^{2(q-2)} & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & \alpha^\mu & \alpha^{2\mu} & \cdots & \alpha^{\mu(q-2)} & 0 \\ \end{array} \right)$$ Let $q-1=p_1^{t_1}\, p_2^{t_2}\,\cdots\, p_n^{t_n}$ where $p_i$, $1\leq i \leq n$, are prime numbers. Suppose that $\beta_i$,$1\leq i \leq n$, be elements of $F_q$ of order $p_i^{t_i}$.

My question: What kind of code the null space of $G_{\beta_i}^{(p_i^{t^i},\mu)}$ generate?

My motivation to ask this question was the section $IV$ of this paper.

Thanks for any suggestions.