Counting the number of codes

chiaroscuro

Hi all.

Let [itex]C[/itex] and [itex]D[/itex] be codes of length n over [itex]\mathbb{F}_q[/itex] of dimension [itex]k[/itex] and [itex]k+l[/itex] respectively.

I want to count the number of codes [itex]D[/itex] such that [itex]C\subseteq D\subseteq (C^\perp\cap D^\perp)[/itex].

I understand that this will involve the Gaussian coefficient. If I'm not mistaken, I think it would be of the form [itex]{A\choose l }_q[/itex] but I can't figure out what [itex]A[/itex] is.

Any help would be much appreciated. Thanks!