#### Volume 20, issue 3 (2016)

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Optimal simplices and codes in projective spaces

### Henry Cohn, Abhinav Kumar and Gregory Minton

Geometry & Topology 20 (2016) 1289–1357
##### 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 $ℍ‘{ℙ}^{2}$ and $27$–point simplices in $\mathbb{O}‘{ℙ}^{2}$, 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 $\mathbb{O}‘{ℙ}^{2}$ that form a maximal system of mutually unbiased bases. This is the last tight code in $\mathbb{O}‘{ℙ}^{2}$ whose existence had been previously conjectured but not resolved.

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##### Keywords
code, design, regular simplex, linear programming bounds, projective space, Grassmannian
##### Mathematical Subject Classification 2010
Primary: 51M16, 52C17
Secondary: 65G20, 49M15

PARI/GP code

##### Publication
Received: 3 February 2014
Revised: 23 July 2015
Accepted: 23 July 2015
Published: 4 July 2016
Proposed: Ian Agol
Seconded: Leonid Polterovich, Walter Neumann
##### Authors
 Henry Cohn Microsoft Research One Memorial Drive Cambridge, MA 02142 United States Abhinav Kumar Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 United States Department of Mathematics Stony Brook University Stony Brook, NY 11794 United States Gregory Minton Microsoft Research One Memorial Drive Cambridge, MA 02142 United States