Optimality of spherical codes via exact semidefinite programming bounds

TL;DR

This paper proves the optimality of certain spherical codes and arrangements using advanced semidefinite programming bounds, including new cases and techniques for exact solutions, with implications for coding theory and sphere packings.

Contribution

It introduces new exact semidefinite programming bounds and techniques to establish the optimality and uniqueness of specific spherical codes and related structures.

Findings

Optimality of spectral embeddings of triangle-free strongly regular graphs.

Optimality of certain mutually unbiased basis arrangements constructed from Kerdock codes.

Universal optimality of 288 points on a sphere in 16 dimensions.

Abstract

We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as are certain mutually unbiased basis arrangements constructed using Kerdock codes in up to $1024$ dimensions (namely, $2^{4k} + 2^{2k+1}$ points in $2^{2k}$ dimensions for $2 \le k \le 5$). As a consequence of the latter, we obtain optimality of the Kerdock binary codes of block length $64$, $256$, and $1024$, as well as uniqueness for block length $64$. We also prove universal optimality for $288$ points on a sphere in $16$ dimensions. To prove these results, we use three-point semidefinite programming bounds, for which only a few sharp cases were known previously. To obtain rigorous results, we develop improved techniques for rounding approximate solutions of semidefinite programs to produce exact optimal solutions.