Subspace Codes Based on Partial Injective Maps of Vector Spaces Over Finite Fields

Subspace codes are widely used in error corrections of random network coding. In this article, subspace codes based on partial injective maps of vector spaces over finite fields are considered. Several bounds of the subspace codes
${(n,M,2b,e)_{q}}$
based on
$e$
-partial injective maps of
$\mathbb {F}_{q}^{(n)}$
are presented. The anticode bound and Ahlswede-Aydinian bound of the subspace codes
${(n,M,2b,e)_{q}}$
are obtained by using the EKR theorem for
$e$
-partial injective maps of
$\mathbb {F}_{q}^{(n)}$
. Finally, we show that the
${(n,M,2b,e)_{q}}$
subspace codes based on
$e$
-partial injective maps of
$\mathbb {F}_{q}^{(n)}$
reach the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in
$I_{e}^{n}$
.