I am interested to know if there is a standard algorithm for generating all the lattice points inside the $d$-dimensional unit ball (with respect to the $\ell_2$-norm).
The brute force approach is to generate the lattice over the $\ell_\infty$ unit ball and then loop through each of the points and remove those with norm strictly greater than $1$. I was wondering if there is a more elegant approach to solving this problem. My initial thoughts are to attempt to cover the $d-1$ dimensional sphere with $n$ points and try and use those points to find out the "limits" that I can then use to generate the lattice coordinate vectors. Then I would define the lattice over the intervals defined by these limits $[a_1,b_1] \times [a_2,b_2] \times … \times [a_d,b_d]$. However, in my mind it is non-trivial to decide how many points to put on the sphere and the algorithm for actually determining the aforementioned limits.
Another approach would be to think of what are the coordinates of the maximum volume polyhedron that can be inscribed in the ball and then somehow constrain them to be on the lattice...Again, this seems to be non trivial. A similar line of thought would be to find the minimum volume polyhedron in which we can include the unit ball by enforcing the constraint that its vertices are lattice points. Then we can just subtract the distance between two lattice points and end up inside the sphere...
Are any of these problems with known solutions/non-trivial to solve?
Thank you in advance.