A representation of Galois dual codes of algebraic geometry codes via Weil differentials

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