  We use a map to quantum error-correcting codes and a subspace projection to
get lower bounds for minimal homological distances in a tensor product of two
chain complexes of vector spaces over a finite field. Homology groups of such a
complex are described by the K\"unneth theorem. We give an explicit expression
for the distances when one of the complexes is a linear map between two spaces.
The codes in the construction, subsystem product codes and their gauge-fixed
variants, generalize several known families of quantum error-correcting codes.
