A representation of Galois dual codes of algebraic geometry codes via Weil differentials
Authors: Li Jiaqi , Ma Liming ,   School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China,   Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
Published: November 2023

Galois dual codes are a generalization of Euclidean dual codes and Hermitian dual codes. We show that the <inline-formula><tex-math id="M910">\begin{document}$ h $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M910.png"/></alternatives></inline-formula>-Galois dual code of an algebraic geometry code <inline-formula><tex-math id="M900">\begin{document}$ C_{ {\cal{L}},F}(D,G) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M900.png"/></alternatives></inline-formula> from function field <inline-formula><tex-math id="M904">\begin{document}$ F/ \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M904.png"/></alternatives></inline-formula> can be represented as an algebraic geometry code <inline-formula><tex-math id="M902">\begin{document}$ C_{\varOmega,F'}(\phi_{h}(D),\phi_{h}(G)) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M902.png"/></alternatives></inline-formula> from an associated function field <inline-formula><tex-math id="M903">\begin{document}$ F'/ \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M903.png"/></alternatives></inline-formula> with an isomorphism <inline-formula><tex-math id="M600">\begin{document}$\phi_{h}:F\rightarrow F'$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M600.png"/></alternatives></inline-formula> satisfying <inline-formula><tex-math id="M700">\begin{document}$ \phi_{h}(a) = a^{p^{e-h}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M700.png"/></alternatives></inline-formula> for all <inline-formula><tex-math id="M800">\begin{document}$ a\in \mathbb{F}_{p^e} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0019_M800.png"/></alternatives></inline-formula>. As an application of this result, we construct a family of <i>h</i>-Galois linear complementary dual maximum distance separable codes (<i>h</i>-Galois LCD MDS codes).