The Nonexistence of Regular Near Octagons with Parameters $(s,t,t_2,t_3)=(2,24,0,8)$

Abstract

Let $\mathcal{S}$ be a regular near octagon with $s+1=3$ points per line, let $t+1$ denote the constant number of lines through a given point of $\mathcal{S}$ and for every two points $x$ and $y$ at distance $i \in \{ 2,3 \}$ from each other, let $t_i+1$ denote the constant number of lines through $y$ containing a (necessarily unique) point at distance $i-1$ from $x$. It is known, using algebraic combinatorial techniques, that $(t_2,t_3,t)$ must be equal to either $(0,0,1)$, $(0,0,4)$, $(0,3,4)$, $(0,8,24)$, $(1,2,3)$, $(2,6,14)$ or $(4,20,84)$. For all but one of these cases, there is a unique example of a regular near octagon known. In this paper, we deal with the existence question for the remaining case. We prove that no regular near octagons with parameters $(s,t,t_2,t_3)=(2,24,0,8)$ can exist.