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Optimal simplices and codes in projective spaces

Henry Cohn, Abhinav Kumar and Gregory Minton

Geometry & Topology 20 (2016) 1289–1357

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 2 and 27–point simplices in O2, 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 n with n 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 O2 that form a maximal system of mutually unbiased bases. This is the last tight code in O2 whose existence had been previously conjectured but not resolved.

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

PARI/GP code

Received: 3 February 2014
Revised: 23 July 2015
Accepted: 23 July 2015
Published: 4 July 2016
Proposed: Ian Agol
Seconded: Leonid Polterovich, Walter Neumann
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