Am I missing something or is your reference about the shortest vector problem rather than the closest vector problem (I think the former is only a spectal case of the latter).

I am totally aware of the fact that this is a hard problem. I'm not hoping for short running time, but it should be possible to compute it in reasonable time for small $n$ and I am looking for implementations of this.

Yoav, is fplll a command in magma or where?

Are you sure that these do what they should? I thought that the LLL algorithm just gives a close vector (which is close enough for many applications) but not necessarily the closest one?

okay, now I am more than happy! Thx!

thanks for your answers Jason! I have a question regarding the second edit. For simplicity, assume that $R$ is a discrete valuation ring with residue field $k$. The following is not apparent to me from the proof: Why can it not be that the maximal open subscheme $U$ of $\mathbb{P}_R^d$, over which $f_*\mathcal{O}_Z$ is locally free, is empty, but $Z\times_R k \to \mathbb{P}_k^d$ is flat? (In that case the fiber over the closed point would be CM, but not the one over the generic point)

Wait, why is this the same as in my question? So Cor. 6.11.3 sais that the set of points $x$ such that $F_x$ is CM is open. But if the base is e.g. a field $K$ then being CM for a $K$-algebra is not the same as being CM as a $K$-module (which is always the case).

Hi Jason, thanks for this answer! Just two quick question: What happens for rings of dimension 1? Is it true that reflexive is equivalent to projective? And do you have a reference for your first statement, namely that the module is reflexive iff its depth is at least two?