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

Abstract: An $(r,M,2\delta;k)_q$ constant--dimension subspace code, $\delta &gt;1$, is a collection $\cal C$ of $(k-1)$--dimensional projective subspaces of ${\rm PG(r-1,q)}$ such that every $(k-\delta)$--dimensional projective subspace of ${\rm PG(r-1,q)}$ is contained in at most a member of $\cal C$. 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 $n &gt; 4$. When $n=4$ a further improvement is provided by constructing an $(8,M,4;4)_q$ constant--dimension subspace code, with $$M = q<sup>{12}+q<sup>2(q<sup>2+1)<sup>2(q<sup>2+q+1)+1$$.