Optimality and uniqueness of the (4,10,1/6) spherical code

TL;DR

This paper demonstrates the optimality and uniqueness of the Petersen code as the (4,10,1/6) spherical code using semidefinite programming bounds, overcoming limitations of linear programming methods.

Contribution

It introduces a novel application of semidefinite programming bounds to establish the uniqueness of a specific spherical code where linear programming fails.

Findings

The Petersen code is the unique (4,10,1/6) spherical code.

Semidefinite programming bounds can prove optimality and uniqueness in cases where linear programming cannot.

The paper provides a new method for analyzing spherical codes with challenging parameters.

Abstract

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code, which consists of the midpoints of the edges of the regular simplex in dimension 4, is the unique (4,10,1/6) spherical code.