Geometric Hyperplanes of the Near Hexagon L_3 times GQ(2, 2)

TL;DR

This paper classifies all geometric hyperplanes of a specific near hexagon, revealing their structure and relation to the Veldkamp space, with potential implications for quantum physics applications.

Contribution

It provides a complete classification of geometric hyperplanes of the near hexagon L_3 times GQ(2, 2), detailing their types and relationships to Veldkamp space.

Findings

Eight types of hyperplanes identified

Total hyperplanes count: 1023

Hyperplanes form two distinct families

Abstract

Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle of order two. There are eight different kinds of them, totalling to 1023 = 2^{10} - 1 = |PG(9, 2)|, and they form two distinct families intricately related with the points and lines of the Veldkamp space of the quadrangle in question.