Optimal simplices and codes in projective spaces

Cohn, Henry; Kumar, Abhinav; Minton, Gregory

doi:10.2140/gt.2016.20.1289

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Optimal simplices and codes in projective spaces

Henry Cohn, Abhinav Kumar and Gregory Minton

Geometry & Topology 20 (2016) 1289–1357

DOI: 10.2140/gt.2016.20.1289

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 $O ‘ ℙ^{2}$ , 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 $n \leq 8$ ; 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 $O ‘ ℙ^{2}$ that form a maximal system of mutually unbiased bases. This is the last tight code in $O ‘ ℙ^{2}$ whose existence had been previously conjectured but not resolved.

Keywords

code, design, regular simplex, linear programming bounds, projective space, Grassmannian

Mathematical Subject Classification 2010

Primary: 51M16, 52C17

Secondary: 65G20, 49M15

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References

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

Volume 20, issue 3 (2016)