Optimality of spherical codes via exact semidefinite programming bounds
<|reference_start|>Optimality of spherical codes via exact semidefinite programming bounds: 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.<|reference_end|>
@article{cohn2024optimality,
    title={Optimality of spherical codes via exact semidefinite programming bounds},
    author={Henry Cohn, David de Laat, Nando Leijenhorst},
    journal={arXiv preprint arXiv:2403.16874},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16874},
    primaryClass={math.MG cs.IT math.IT math.OC}
}