On the uniqueness of a generalized quadrangle of order (4,16)

TL;DR

The paper resolves the uniqueness question for generalized quadrangles of order $(4,16)$ by showing any such structure must contain a 3-regular triad, which constrains the configuration to be isomorphic to the known orthogonal quadrangle $Q(5,4)$. The method translates local incidence relations into a $3$-$(17,5,3)$ design on a 17-point set and analyzes block multiplicities with respect to the GQ axioms, ultimately forcing a single realization. Consequently, there is a unique $ ext{GQ}(4,16)$ up to isomorphism, and the associated strongly regular graph with parameters $(325,68,3,17)$ is likewise unique and geometric. This result advances the classification of generalized quadrangles and their connections to designs and strongly regular graphs, confirming the exclusivity of the known example in this parameter set.

Abstract

In the manuscript [v3], we prove the uniqueness of a generalized quadrangle of order (4,16).

Theorems & Definitions (26)

• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Remark 3.5
• Lemma 3.6
• proof
• Lemma 3.7