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

Abstract

Let \begin{document}$ G_2(q) $\end{document} be a Chevalley group of type \begin{document}$ G_2 $\end{document} over a finite field \begin{document}$ \mathbb{F}_q $\end{document}. Considering the \begin{document}$ G_2(q) $\end{document}-primitive action of rank \begin{document}$ 3 $\end{document} on the set of \begin{document}$ \frac{q^3(q^3-1)}{2} $\end{document} hyperplanes of type \begin{document}$ O_{6}^{-}(q) $\end{document} in the \begin{document}$ 7 $\end{document}-dimensional orthogonal space \begin{document}$ {{\rm{PG}}}(7, q) $\end{document}, 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.