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}_۲(۴), \mathcal{P}_۲(۵), \mathcal{P}_۲(۶) and \mathcal{P}_۳(۴). It is shown that the groups ۱, \mathbb{Z}_۲, \mathbb{Z}_۲\times \mathbb{Z}_۲, S_۳, S_۴ and ۱, \mathbb{Z}_۲, \mathbb{Z}_۲\times \mathbb{Z}_۲, S_۳, D_۸, S_۳\times \mathbb{Z}_۲, S_۴, S_۵ appears between these sets in \mathcal{P}_۲(۴) and \mathcal{P}_۲(۵), respectively. Moreover, the groups ۱, \mathbb{Z}_۲, \mathbb{Z}_۲\times \mathbb{Z}_۲, S_۳, D_۸, \mathbb{Z}_۲\times \mathbb{Z}_۲ \times \mathbb{Z}_۲, S_۳\times \mathbb{Z}_۲, D_۸\times \mathbb{Z}_۲, S_۴, S_۳\times S_۳, S_۴\times \mathbb{Z}_۲, (S_۳\times S_۳):۲, S_۵, S_۶ and ۱, \mathbb{Z}_۲, \mathbb{Z}_۲\times \mathbb{Z}_۲, S_۳, D_۸, S_۴ appears between these sets in \mathcal{P}_۲(۶) and \mathcal{P}_۳(۴), respectively.

کلیدواژه ها:

subspace code ، subspace distance ، pivot vector