TITLE: The Number of Short Vectors in a Lattice
QUESTION [11 upvotes]: Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the number of lattice vectors $v$ with $|v| \leq c$ without actually enumerating these vectors?  The basis $v_1,\ldots, v_m$ of the lattice can be assumed to be LLL reduced.
Also asked at: https://cstheory.stackexchange.com/questions/7488/the-number-of-short-vectors-in-a-lattice

REPLY [9 votes]: For this problem one typically employs the so-called Gaussian heuristic: 

if $K$ is a measurable subset of the
  span of the $n$-dimensional lattice
  $L$, then $| K \cap L | \approx
> \mbox{vol}(K)/\det(L)$.

In particular, the case for $K$ a ball is used in some (enumerative) SVP/CVP solvers. See $\S 5$ of
"Algorithms for the shortest and closest lattice vector problems"
by Hanrot, Pujol and Stehlé.