An Alternative Approach to the Classification of Regular Near Hexagons with Parameters \((s, t,t_2) = (2, 11, 1)\)

Abstract

Based on some results of Shult and Yanushka [7], Brouwer [1] proved that there exists a unique regular near hexagon with parameters \((s,t,t_2) = (2,11,1)\), namely the one related to the extended ternary Golay code. His proof relies on the uniqueness of the Witt design \(S(5,6,12)\), Pless’s characterization of the extended ternary Golay code \(G_{12}\), and some properties of \(S(5,6,12)\) and \(G_{12}\). It is possible to avoid all this machinery and provide an alternative, more elementary and self-contained proof for the uniqueness. The author recently observed that such an alternative proof is implicit in the literature, obtainable by combining results from [1], [4], and [7]. This survey paper aims to bring this fact to the attention of the mathematical community. We describe the relevant parts of the above papers for this alternative proof of classification. Additionally, we prove several extra facts not explicitly contained in [1], [4], or [7]. This paper can also be seen as an addendum to Section 6.5 of [3], where the uniqueness of the near hexagon was not proved.