0    & 1    & 0     & w^3 & 0     & 0   &w^3& w  & 1 \\
0 & 1 & 0 & 0 & 1      &w^5&w^4&w^4 & 1      & w^5 & 0   &1     &0    &0  \\ 
0 & 0 &1  & 0 &w^2  &1     &0     &0     &0        &1      &w^6&w^2&0   &w^6\\
0 & 0 &0  &1  &0       & w   & 1     &1    &0        &0      &1     &w^3&w   &0\\
\end{pmatrix}. \end{aligned}$$ For simplicity of notation, let us denote $w,w^2,\dots,w^6$ with $2,3,\dots,7$, respectively. With this notation, the above parity-check matrix can be written simply as $[g]^7+P_2^7=[4130]^7+0501^7.$

Consider $\F_9=\{ 0,1,\beta,\beta^2,\dots,\beta^7\}$, where $\beta$ is a root of $x^2+2x+2\in\F_3[x]$. We denote $0,1,\beta,\beta^2,\dots,\beta^7$ by $0,1,2,3,\dots,8$, respectively. We set that $g(x)=x^4+(1+8x+5x^2+4x^3)=x^4-(5+4x+x^2+8x^3) \in\F_9[x;\theta]$ divides $x^5-2$ with $\theta(a)=a^3$ for every $a\in\F_9$. Let $\mathcal{T}$ be the semi-linear map defined by $\theta$ and $g(x)$. Consider the following three points $P_1$, $P_2$ and $P_3$ of $\P^3(\F_9)$: $$P_1=[1:0:0:0], P_2=[1:6:8:1], P_3=[4:7:4:1].$$ Then, under $\mathcal{T}$, the orbit of $P_1$ is of length $5$ and the orbits of $P_2$ and $P_3$ are both of length $10$. Thus we deduce that $$G=[4518]^5+1681^{10}+4741^{10}$$ generate a SQT $[25,4,19]_9$-code. By comparing the distance of this code with that in the database of MAGMA, we can observe that it reaches the best known 