  For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is
the $\mathbb{Z}$-lattice in $\mathbb{R}^d$ generated by the Vandermonde matrix
associated to the roots of the $d$-dimensional Chebyshev polynomial. It is
important to enumerate the points from the Chebyshev-Frolov lattices in
axis-parallel boxes when $d = 2^n$ for a non-negative integer $n$, since the
points are used for the nodes of Frolov's cubature formula, which achieves the
optimal rate of convergence for many spaces of functions with bounded mixed
derivatives and compact support. The existing enumeration algorithm for such
points by Kacwin, Oettershagen and Ullrich is efficient up to dimension $d=16$.
In this paper we suggest a new enumeration algorithm of such points for
$d=2^n$, efficient up to $d=32$.
