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We give an inequality on the packing of vectors/lines in quaternionic Hilbertspace $\Hd$, which generalises those of Sidelnikov and Welch for unit vectorsin $\Rd$ and $\Cd$. This has a parameter $t$, and depends only on the vectorsup to projective unitary equivalence. The sequences of vectors in${\mathbb{F}}^d={\mathbb{R}}^d,{\mathbb{C}}^d,{\mathbb{H}}^d$ that giveequality, which we call spherical $(t,t)$-designs, are seen to satisfy acubature rule on the unit sphere in ${\mathbb{F}}^d$ for a suitable polynomialspace $\Hom_{\Fd}(t,t)$. Using this, we show that the projective spherical$t$-designs on the Delsarte spaces $\FF P^{d-1}$ coincide with the spherical$(t,t)$-designs of unit vectors in ${\mathbb{F}}^d$. We then explore a numberof examples in quaternionic space. The unitarily invariant polynomial space${\mathop{\rm Hom}\nolimits}_{{\mathbb{H}}^d}(t,t)$ and the inner product thatwe define on it so the reproducing kernel has a simple form are of independentinterest.

A variational characterisation of projective spherical designs over the quaternions