A representation of Galois dual codes of algebraic geometry codes via Weil differentials
Jiaqi Li, Liming Ma · JUSTC · 2023
Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the \begin{document}$ h $\end{document} -Galois dual code of an algebraic geometry code \begin{document}$ C_{ {\cal{L}},F}(D,G) $\end{document} from function field \begin{document}$ F/ \mathbb{F}_{p^e} $\end{document} can be represented as an algebraic geometry code \begin{document}$ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $\end{document} from an associated function field \begin{document}$ F'/ \mathbb{F}_{p^e} $\end{document} with an isomorphism \begin{document}$\phi_{h}:F\rightarrow F'$\end{document} satisfying \begin{document}$ \phi_{h}(a) = a^{p^{e-h}} $\end{document} for all \begin{document}$ a\in \mathbb{F}_{p^e} $\end{document} . As an application of this result, we construct a family of h-Galois linear complementary dual maximum distance separable codes (h-Galois LCD MDS codes).