Structured spherical codes with asymptotically optimal distance distributions

We introduce a new geometric construction of cyclic group codes in odd-dimensional spaces formed by intersecting even-dimensional constant curvature curves with hyperplanes of one less dimension. This allows us to recast the cyclic group code as a uniform sampling of a constant curvature curve whereby the design of the constant curvature curve controls code performance. Using a tool from knot theory known as the circumradius function, we derive properties of cyclic group codes from properties of the constant curvature curve passing through every point of the spherical code. By relating the distribution of the squared circumradius function of the connecting curve to the distribution of the pairwise distances of the cyclic group code, we show that the distance spectrum of cyclic group codes achieves optimality in the sense of matching the random spherical code distance distribution bound as the block length grows large.