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

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.

💡 Research Summary

The paper tackles a long‑standing open problem in spherical coding: proving both optimality and uniqueness of a (4,10,1/6) spherical code. A spherical code consists of N points on the unit sphere S^{n‑1} such that the inner product between any two distinct points does not exceed a prescribed value t. The parameters (n,N,t) are traditionally denoted as (n,N,t). For most parameter sets, the Delsarte‑Goethals‑Seidel linear programming (LP) bound—derived from positive‑definite polynomial techniques—provides a sharp upper bound on N and often yields uniqueness results. However, the specific case (4,10,1/6) lies outside the reach of the LP method; the LP bound fails to prove that N cannot exceed 10, leaving open the possibility of a better configuration.

To overcome this limitation, the authors employ semidefinite programming (SDP) bounds, which generalize the LP approach by allowing matrix‑valued positive‑definite constraints. The core idea is to construct two auxiliary polynomials p(t) and q(t) of degrees 2 and 3, respectively, whose coefficients are chosen so that the associated Gram matrix G(p,q) captures the inner‑product structure of any candidate code. If G(p,q) can be shown to be positive semidefinite, then an SDP bound follows, yielding an explicit upper limit on the number of points N for the given t.

The authors formulate an SDP where the decision variables are the coefficients of p and q, together with a set of matrix variables representing the Gram matrix. They solve this SDP numerically using high‑precision solvers such as SDPA and MOSEK, employing 128‑bit floating‑point arithmetic to control rounding errors. The numerical solution confirms that G(p,q) is indeed positive semidefinite, which translates into the rigorous inequality N ≤ 10 for any (4,10,1/6) code. Since the Petersen code—obtained by taking the midpoints of the edges of a regular 4‑simplex—realizes N = 10 and achieves the inner‑product value 1/6, the SDP bound proves that the Petersen code is optimal.

Beyond optimality, the paper establishes uniqueness. The SDP optimum is shown to be attained at a unique extreme point of the feasible region. By examining the Lagrange multipliers associated with the optimal solution, the authors reveal that these multipliers exhibit the symmetry of the alternating group A₅, which is precisely the symmetry group of the Petersen code. Consequently, any other configuration achieving the same parameters would have to share this symmetry, forcing it to be isometric to the Petersen code. Hence the Petersen code is the sole (4,10,1/6) spherical code up to orthogonal transformations.

The authors also address the reliability of the numerical evidence. They provide rigorous error bounds, demonstrating that the computed solution deviates from the exact SDP optimum by less than 10⁻¹² in absolute terms. To ensure reproducibility, they release the source code and data used for the SDP computations, allowing independent verification.

In conclusion, the paper demonstrates that semidefinite programming can fill the gaps left by linear programming in spherical code theory. By applying SDP bounds, the authors not only prove that the Petersen code is optimal for the (4,10,1/6) parameters but also that it is uniquely optimal. This work suggests a broader applicability of SDP techniques to other high‑dimensional, high‑density coding problems where traditional LP bounds are insufficient, and it opens avenues for future research into more complex configurations and refined SDP formulations.