Nonexistence of some ternary linear codes with minimum weight -2 modulo 9

<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id="M3">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id="M9">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id="M10">\begin{document}$ n_3(6,d) $\end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id="M11">\begin{document}$ d $\end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>

On some Ternary Linear Codes and Designs obtained from the Projective Special Linear Goup PSL3(4)

Let (G, ∗) be a group and X any set, an action of a group G on X, denoted as G×X → X, (g, x) 7→ g.x, assigns to each element in G a transformation of X that is compatible with the group structure of G. If G has a subgroup H then there is a transitive group action of G on the set (G/H) of the right co-sets of H by right multiplication. A representation of a group G on a vector space V carries the dimension of the vector space. Now, given a field F and a finite group G, there is a bijective correspondence between the representations of G on the finitedimensional F-vector spaces and finitely generated FG-modules. We use the FG -modules to construct linear ternary codes and combinatorial designs from the permutation representations of the group L3(4). We investigate the properties and parameters of these codes and designs. We further obtain the lattice structures of the sub-modules and compare these ternary codes with the binary codes constructed from the same group.