# Finding lattice vector with entries of low height

Hello

Given a lattice $L \subseteq \mathbb{Q}^n$ we can define the height of a given element $v = (v_1,\ldots,v_n)\in L$ to be $$\operatorname{ht}(v) = \sum_{i=1}^n \operatorname{ht}(v_i)$$ where as usual the height of an individual rational number $p/q$ is given by $$\operatorname{ht}(p/q) := \max(|p|,|q|)$$ provided $p/q$ is written in lowest terms. Does there exist an algorithm which, given a basis for the lattice $L$, will return an element of $L$ of low height.

More informally I am interested in knowing whether there exists a systematic way of producing elements in a rational lattice with relatively simple entries (i.e. the entries are suppose to be rational numbers of small height compared to the entries of the basis elements). Ideally I would of course like to find the element with simplest possible entries, but any thoughts on the matter/partial results would be of interest.