Efficient methods to build spherical designs in odd dimensions
Efficient lifts into odd dimensions of Spherical $5$- and $7$-designs with Balanced Halves
Information Theory
Summary
Spherical designs are special sets of points spread evenly on the surface of a sphere, useful in many areas like physics and computer graphics. The paper presents new ways to create these sets in odd-numbered dimensions, improving how efficiently they can be constructed. The authors do this by cleverly using known designs from even dimensions and extending them into more layers. Their constructions keep the number of points needed low, which is important for practical use. They also connect their results to a well-known mathematical conjecture to show even better outcomes if it holds true.
What this means in practice
- •For 3d graphics developers: Create more efficient point distributions for rendering effects on spherical objects in odd dimensions, improving visual quality with fewer points.
- •For signal processing engineers: Design sampling schemes on spherical domains in odd dimensions that reduce the number of sensors needed while preserving accuracy.
Authors
Konstantin Delchev, Boryan Yordanov
Abstract
We propose constructions of spherical $5$- and $7$-designs in odd dimensions by lifting balanced halves of even-dimensional designs to several layers of suitable latitudes. For every odd dimension, we obtain an explicit spherical $5$-design with at most $60(d^2-1)$ points, attaining the optimal quadratic order in the dimension. Assuming Zauner's conjecture, this bound improves to $7.5(d-1)^2$. For every odd $d \geq 9$, we also construct a spherical $7$-design with $110M$ points from a complex projective $3$-design with $M$ points in $\mathbb{CP}^{(d-3)/2}$.