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2016 Optimal simplices and codes in projective spaces
Henry Cohn, Abhinav Kumar, Gregory Minton
Geom. Topol. 20(3): 1289-1357 (2016). 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 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.

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Henry Cohn. Abhinav Kumar. Gregory Minton. "Optimal simplices and codes in projective spaces." Geom. Topol. 20 (3) 1289 - 1357, 2016. https://doi.org/10.2140/gt.2016.20.1289

Information

Received: 3 February 2014; Revised: 23 July 2015; Accepted: 23 July 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1353.90084
MathSciNet: MR3523059
Digital Object Identifier: 10.2140/gt.2016.20.1289

Subjects:
Primary: 51M16, 52C17
Secondary: 49M15, 65G20

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.20 • No. 3 • 2016
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