  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.
