Constructive Spherical Codes by Hopf Foliations

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
$O(n \log n)$
.