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 $piin s_n$, $nu(u)$ 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 $mathcal{g}_{u,v}={piin s_nmid d_s(u,v)=d_h(nu(pi u),nu(pi v))}$ for any $u,vinmathcal{p}_q(n)$ 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 $1$, $mathbb{z}_2$, $mathbb{z}_2times mathbb{z}_2$, $s_3$, $s_4$ and $1$, $mathbb{z}_2$, $mathbb{z}_2times mathbb{z}_2$, $s_3$, $d_8$, $s_3times mathbb{z}_2$, $s_4$, $s_5$ appears between these sets in $mathcal{p}_2(4)$ and $mathcal{p}_2(5)$, respectively. moreover, the groups $1$, $mathbb{z}_2$, $mathbb{z}_2times mathbb{z}_2$, $s_3$, $d_8$, $mathbb{z}_2times mathbb{z}_2 times mathbb{z}_2$, $s_3times mathbb{z}_2$, $d_8times mathbb{z}_2$, $s_4$, $s_3times s_3$, $s_4times mathbb{z}_2$, $(s_3times s_3)$:$2$, $s_5$, $s_6$ and $1$, $mathbb{z}_2$, $mathbb{z}_2times mathbb{z}_2$, $s_3$, $d_8$, $s_4$ appears between these sets in $mathcal{p}_2(6)$ and $mathcal{p}_3(4)$, respectively.