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.

On the subspace distance of the subspace codes Keywords:

subspace code , subspace distance , pivot vector