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.

翻译：我们证明了所有已知无三角形强正则图的谱嵌入是最优球面码（新情形包括20维空间中的56个点、21维空间中的50个点以及21维空间中的77个点），同时，利用Kerdock码构建的某些互偏基排列（具体而言，对于$2 \le k \le 5$，在$2^{2k}$维空间中包含$2^{4k} + 2^{2k+1}$个点）也具有最优性。由此推论，我们获得了码长分别为64、256和1024的Kerdock二元码的最优性，以及码长为64时的唯一性。此外，我们还证明了16维球面上288个点集的普适最优性。为证明这些结论，我们采用了三点半定规划界，此前仅有少数尖锐情形已知该技术。为确保结果的严谨性，我们发展了改进技术来舍入半定规划的近似解，从而精确获得最优解。