Lattices and shortest vector
In my book about Lattice basis reduction I found the following:
Gaussian Heuristic: for big enough, there are approximately lattice points in of norm , where denotes the volume of the m-dimensional unit ball.
Corollary: for a lattice of rank the shortest nonzero lattice vector has norm at most approximately .
Then they give this example:
Given , and > with large. Small values for the linear form with corresponds to lattices
How large should be taken to compute with and minimal?
Their answer is:
The Gaussian heuristic with expects solutions of size approximately , with approximately .
How do they come up with ?
Using the above Corallary I have that the shortest nonzero lattice vector has norm at most approximately
Then since we have that the norm of .
So and thus is approximately ?
Am I doing this right? I hope someone can explain where the is coming from.