New Constant-Dimension Subspace Codes from Maximum Rank Distance Codes

The main problem of constant-dimension subspace coding is to determine the maximal possible size
${\mathbf{A}}_{q}(n,d,k)$
of a set of
$k$
-dimensional subspaces in
${\mathbf{F}}_{q}^{n}$
such that the subspace distance satisfies
$d(U,V) \geq d$
for any two different subspaces
$U$
and
$V$
in this set. In this paper, we give a direct construction of constant-dimension subspace codes from two parallel versions of maximum rank-distance codes. The problem about the sizes of our constructed constant-dimension subspace codes is transformed into finding a suitable sufficient condition to restrict number of the roots of
$L_{1}(L_{2}(x))-x$
where
$L_{1}$
and
$L_{2}$
are
$q$
-polynomials over the extension field
${\mathbf{F}}_{q^{n}}$
. New lower bounds for
${\mathbf{A}}_{q}(4k,2k,2k)$
,
${\mathbf{A}}_{q}(4k+2,2k,2k+1)$
, and
${\mathbf{A}}_{q}(4k+2,2(k-1),2k+1)$
are presented. Many new constant-dimension subspace codes better than previously best known codes with small parameters are constructed.