Some results on linear subspace codes

Abstract

The notion of linearity in projective spaces was defined by Etzion and Vardy (2008) [3]. In this paper, we have obtained some results on linear subspace codes. We have proved that if in a linear subspace code C in the projective space $P_q\left(n\right)$, the number of one-dimensional subspaces is $n-1$, then the cardinality of C is $2^{n-1}$; and if the number of the one-dimensional subspaces in C is $n-2$ and the ambient space does not belong to C, then the cardinality of C is $2^{n-2}$. We have also studied complementary linear subspace codes. An example has been given to show that a complement function can exist on a non-distributive sublattice of the projective lattice. We have also proved that a non-distributive sublattice of the projective lattice cannot be a linear subspace code.