  We present a new systematic approach to constructing spherical codes in
dimensions $2^k$, based on Hopf foliations. Using the fact that a sphere
$S^{2n-1}$ is foliated by manifolds $S_{\cos\eta}^{n-1} \times
S_{\sin\eta}^{n-1}$, $\eta\in[0,\pi/2]$, we distribute points in dimension
$2^k$ via a recursive algorithm from a basic construction in $\mathbb{R}^4$.
Our procedure outperforms some current constructive methods in several
small-distance regimes and constitutes a compromise between achieving a large
number of codewords for a minimum given distance and effective constructiveness
with low encoding computational cost. Bounds for the asymptotic density are
derived and compared with other constructions. The encoding process has storage
complexity $O(n)$ and time complexity $O(n \log n)$. We also propose a
sub-optimal decoding procedure, which does not require storing the codebook and
has time complexity $O(n \log n)$.
