On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ G_2(q) $\end{document}</tex-math></inline-formula> be a Chevalley group of type <inline-formula><tex-math id="M3">\begin{document}$ G_2 $\end{document}</tex-math></inline-formula> over a finite field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula>. Considering the <inline-formula><tex-math id="M5">\begin{document}$ G_2(q) $\end{document}</tex-math></inline-formula>-primitive action of rank <inline-formula><tex-math id="M6">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> on the set of <inline-formula><tex-math id="M7">\begin{document}$ \frac{q^3(q^3-1)}{2} $\end{document}</tex-math></inline-formula> hyperplanes of type <inline-formula><tex-math id="M8">\begin{document}$ O_{6}^{-}(q) $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M9">\begin{document}$ 7 $\end{document}</tex-math></inline-formula>-dimensional orthogonal space <inline-formula><tex-math id="M10">\begin{document}$ {{\rm{PG}}}(7, q) $\end{document}</tex-math></inline-formula>, we study the designs, codes, and some related geometric structures. We obtained the main parameters of the codes, the full automorphism groups of these structures, and geometric descriptions of the classes of minimum weight codewords.</p>