Subspace codes in PG(2n-1,q)

An $(r,M,2\delta ;k{)}_q$ constant–dimension subspace code, $\delta >1$, is a collection $\mathcal{C}$ of $(k-1)$–dimensional projective subspaces of ${\rm PG(r-1,q)}$suchthatevery$(k-\delta)$--dimensionalprojectivesubspaceof${\rm PG(r-1,q)}$iscontainedinatmostamemberof$\cal C$. Constant–dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang, where they presented an application of such codes for error-correction in random network coding. Here a $(2n,M,4;n{)}_q$ constant–dimension subspace code is constructed, for every $n\ge 4$. The size of our codes is considerably larger than all known constructions so far, whenever $n>4$. When $n=4$ a further improvement is provided by constructing an $(8,M,4;4{)}_q$ constant–dimension subspace code, with $M = q^{12}+q^2(q^2+1)^2(q^2+q+1)+1$.