full rank designs over Z_m

[Brendan McKay]

In the case of m not prime, one has to be cautious because several
equivalences from the prime case do not hold.  In particular,
being linearly independent is different from having a full-sized
submatrix (in your case, n*n) of non-zero determinant.

I wrote a paper long ago with Richard Brent that counts integer
matrices with given rank defined by the submatrix-determinant method:
   http://cs.anu.edu.au/~bdm/papers/BrentMcKayZm.pdf
I don't see how it applies to your problem easily, but maybe some
of the ideas are useful.

Also, regardless of your problem, that paper seems to provide any
number of sequences not in OEIS (so this is an invitation for some
enterprising oeiser to jump in).  The paper refers to probabilities
but of course you just need to multiply by the total number of
matrices to turn it into a count.

Brendan.


* Max Alekseyev <maxale at gmail.com> [070725 13:49]:
> SeqFans,
> 
> Does the following sounds familiar to anybody?
> 
> Fix an integer m>=2. For any positive integer n consider a matrix k x
> n over Z_m such that any n rows (out of k) are linearly independent.
> Define a(n) as the maximum possible k.
> 
> >From practical point of view such construction gives rise to a number
> of sequences for different m. Are they in OEIS?
> 
> I'm also interested in the theory behind. I believe such matrices
> should have been studied.
> 
> Thanks,
> Max