Optimal simplices and codes in projective spaces

Abstract

We find many tight codes in compact spaces, in other words, optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective plane. The most noteworthy cases are $15$–point simplices in $ℍ‘{\mathbb{P}}^2$ and $27$–point simplices in $\mathbb{O}‘{\mathbb{P}}^2$, both of which are the largest simplices and the smallest $2$–designs possible in their respective spaces. These codes are all universally optimal, by a theorem of Cohn and Kumar. We also show the existence of several positive-dimensional families of simplices in the Grassmannians of subspaces of ${ℝ}^n$ with $n\le 8$; close numerical approximations to these families had been found by Conway, Hardin and Sloane, but no proof of existence was known. Our existence proofs are computer-assisted, and the main tool is a variant of the Newton–Kantorovich theorem. This effective implicit function theorem shows, in favorable conditions, that every approximate solution to a set of polynomial equations has a nearby exact solution. Finally, we also exhibit a few explicit codes, including a configuration of $39$ points in $\mathbb{O}‘{\mathbb{P}}^2$ that form a maximal system of mutually unbiased bases. This is the last tight code in $\mathbb{O}‘{\mathbb{P}}^2$ whose existence had been previously conjectured but not resolved.