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 k 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.