Optimal simplices and codes in projective spaces

TL;DR

This paper demonstrates the existence of optimal, universally optimal simplices and codes in various projective spaces, including quaternionic and octonionic spaces, using computer-assisted proofs and the Newton-Kantorovich theorem.

Contribution

It introduces new families of tight simplices in quaternionic and octonionic projective spaces and proves their existence with computer-assisted methods, expanding the understanding of optimal codes.

Findings

Existence of 15-point simplices in HP^2 and 27-point simplices in OP^2.

Construction of positive-dimensional families of simplices in Grassmannians for n <= 8.

Explicit example of 39 points in OP^2 forming a maximal mutually unbiased bases system.

Abstract

We find many tight codes in compact spaces, i.e., 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 HP^2 and 27-point simplices in OP^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 <= 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 OP^2 that form a maximal system of mutually unbiased bases. This is the last tight code in OP^2 whose existence had been previously conjectured but not resolved.