## Finding basis with short integer basis-vectors spanning a given vectorspace [closed]

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Finding lattice with short basis-vectors containing given lattice

Hello

While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$ I am would like to find $w_1, \ldots w_n \in \mathbb{Z}^m$ such that

$$1.) \ \ \ \ \ \ \ \mathbb{Z}v_1+\dots +\mathbb{Z}v_n \ \subseteq \mathbb{Z}w_1 + \dots + \mathbb{Z}w_n$$

$$2.) \ \ \ \ \ \ \ \ |w_i|^2 \ \text{is small/ small as possible/ a lot smaller that |v|^2}$$

$$3.) \ \ \ \ \ \ \ \ \ \text{the vector space spanned by {v_i} and {w_i} are identical}$$

In other words if each $v_i$ satisfies that $v_{i,1}\alpha_1 + \ldots + v_{i,m}\alpha_m =0$ for a given fixed collection of $\alpha_1,\dots, \alpha_m \in \mathbb{R}$, then the same should be true for the $w_i$.
If I wanted the lattices spanned by $v_i$ and $w_i$ to be identical LLL would clearly be the natural approach, but since I am not require this, this seems to not make my $v_i$ nearly as small as I can achieve under these weaker conditions. Does there exists an algorithm, approach, idea which could come up with a such a basis $v_i$ given the $w_i$.

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How is this different from your question mathoverflow.net/questions/85846/… ? – Yemon Choi Jan 16 2012 at 22:33