On the subspace distance of the subspace codes

Let $\mathcal{P}_q(n)$ be the set of all subspaces in the vector space $\mathbb{F}_q^n$. There is a subspace distance $d_S(U,V)$ between any two subspaces $U$ and $V$. A subspace code is also a subset of $\mathcal{P}_q(n)$. It is known that $d_S(U,V)\geq d_H(\nu(\pi U),\nu(\pi V))$, where $\pi\in S_n$, $\nu(U)$ denotes the pivot vector of $E(U)$ and $E(U)$ is the reduced row echelon form of the generator matrix of $U$. In this paper, we show that if $E(U)$ and $E(V)$ have at most one non-zero entry in each rows and each columns then the equality holds. Moreover, we introduce the sets $\mathcal{G}_{U,V}=\{\pi\in S_n\mid d_S(U,V)=d_H(\nu(\pi U),\nu(\pi V))\}$ for any $U,V\in\mathcal{P}_q(n)$ and examine them in the spaces $\mathcal{P}_2(4)$, $\mathcal{P}_2(5)$, $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$. It is shown that the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $S_4$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_3\times \mathbb{Z}_2$, $S_4$, $S_5$ appears between these sets in $\mathcal{P}_2(4)$ and $\mathcal{P}_2(5)$, respectively. Moreover, the groups $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}_2$, $S_3\times \mathbb{Z}_2$, $D_8\times \mathbb{Z}_2$, $S_4$, $S_3\times S_3$, $S_4\times \mathbb{Z}_2$, $(S_3\times S_3)$:$2$, $S_5$, $S_6$ and $1$, $\mathbb{Z}_2$, $\mathbb{Z}_2\times \mathbb{Z}_2$, $S_3$, $D_8$, $S_4$ appears between these sets in $\mathcal{P}_2(6)$ and $\mathcal{P}_3(4)$, respectively.