Algorithms to generate random orthogonal basis for given Lattice

Suppose I want to generate a $n$-dimentional (random) Lattice, and then output a list of all orthogonal vectors of length $d$. What are the possible algorithms to do this, in poly-time?

one way certainly, is to output the weighted orthonormal vectors, $d.e_i$, where $e_i = \{0, 0, \cdots, i, \cdots, 0, 0\}^T$, but let's say I want my orthogonal vectors to have more structure, and (possibly) random.

I would highly appreciate references, if possible.

• What do you mean by "more structure"? How about generate a random vector in $n$-dimensional space, and then deal with the orthogonal complement space recursively. – Willard Zhan Mar 15 '15 at 13:09
• Thanks, I thought about it, but do you know any concrete algorithm/paper that talks about it? – Subhayan Mar 15 '15 at 13:33