In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.

翻译：本文研究在含 $q$ 个元素的有限域 $\mathbb{F}_q$ 上构造等对偶代数几何（AG）码的问题。给定函数域的有限可分扩张 $\mathcal{M}/\mathcal{F}$ 以及定义在 $\mathcal{F}$ 上的等对偶 AG 码 $\mathcal{C}$，我们提出一种通用方法，在相关不同指数奇偶性满足一定假设的条件下，将码 $\mathcal{C}$ 提升为定义在 $\mathcal{M}$ 上的另一个等对偶 AG 码 $\tilde{\mathcal{C}}$。我们应用该方法，将有理函数域上的等对偶 AG 码提升到初等阿贝尔 $p$-扩张上，例如由 Hermitian、Suzuki 函数域定义的极大函数域，以及由 $GGS$ 函数域覆盖的扩张。我们还获得了定义在分圆扩张上的长二元和三元等对偶 AG 码。