On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $
Tung Le, Bernardo Gabriel Rodrigues · Advances in Mathematics of Communications · 2022
Let $ G_2(q) $ be a Chevalley group of type $ G_2 $ over a finite field $ \mathbb{F}_q $. Considering the $ G_2(q) $-primitive action of rank $ 3 $ on the set of $ \frac{q^3(q^3-1)}{2} $ hyperplanes of type $ O_{6}^{-}(q) $ in the $ 7 $-dimensional orthogonal space $ {{\rm{PG}}}(7, q) $, 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.