yes, but I'm allowing any combination of $(U,V)$ in any order; here $V$ is a permutation of order $n$. So it's equivalence under a larger group containing $GL_k({\mathbb F}_2)$ and $S_n$ as subgroups. (general linear group and symmetric group). Two matrices equivalent under these operations generate equivalent codes (hence the coding theory tag).

I'm allowing these two types of operations : the first is multiply by invertible matrix $U$ : $ G \to U G = [I_k \mid P]$ and the second is a column permutations of $UG$ to make $P$ orthogonal (if possible).

it seems that both puncturing and shortening can only break cycles so I would think $N_{four}' \leq N_{four}$...but I might be overlooking something.

as a final comment, the paper by Morris is as complete as can be; I'm now able to get explicit matrix solutions for any $n,m$.

@DenisSerre,Arul; At this point I'm interested in the question for its own sake; it did come up in some calculations I was making (in optics); the closest thing to these is this (en.wikipedia.org/wiki/Hong%E2%80%93Ou%E2%80%93Mandel_effect‌​), but the connection (if any) is highly speculative

thanks for the references; glad to see that someone has looked at this before. I'll go through the references in more detail, but I think I can see how defining $M_i M_j=\omega M_j M_i$ with $\omega^n=1$ would lead to solutions; although it might be a subset of all possible solutions. Still, there's plenty to work with here...

@Arul, It looks like they use orthogonal designs to define codes. Their examples would have provided solutions for the $n=2$ case if the orthogonal designs are also symmetric $A^T=A$ in their notation which doesn't look like it's the case. Interesting connection though.