On a spherical code in the space of spherical harmonics

In this short note we propose a new method for construction new nice arrangement on the sphere $S^d$ using the spaces of spherical harmonic.

💡 Research Summary

In this short note the authors introduce a novel construction paradigm for spherical codes on the unit sphere $S^{d}$ by exploiting the linear spaces of spherical harmonics. The central idea is to map each point $x\in S^{d}$ to a high‑dimensional Euclidean vector $\Phi_{k}(x)$ whose coordinates are the values of an orthonormal basis of the harmonic space $\mathcal{H}{k}(S^{d})$ of a fixed degree $k$. Because the spherical harmonics are eigenfunctions of the Laplace–Beltrami operator, the mapping $\Phi{k}$ is equivariant with respect to the full rotation group $O(d+1)$ and preserves the norm: $|\Phi_{k}(x)|$ is constant for all $x$. Moreover, the inner product between two such vectors can be expressed in closed form as a Gegenbauer (ultraspherical) polynomial: \