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I have two linear maps $X: \mathcal{L}(G) \rightarrow \mathbb{F}_q^k$ and $Y: \mathcal{L}(2G) \rightarrow \mathbb{F}_{q^n}$, where $G$ is a divisor of the rational function field $F_q(x)$ over $\mathbb{F}_q$ and $\mathcal{L}(G)$ is the Riemann-Roch space of $G$. Currently, I am stuck on computing the inverse maps $X^{-1}$ and $Y^{-1}$. I know I need to use the bases of the two Riemann-Roch spaces and solve linear combinations but I am unsure on exactly how to do this. Any help would be much appreciated.

Edit:
$X$ is defined as follows: For a function $f \in \mathcal{L}(G)$, $f(P_i)$ is computed on $k$ rational places $P_1,..,P_k \in F_q(x)$.

$Y$ is defined as follows: For a function $f \in \mathcal{L}(2G)$, $f(P)$ is computed on a place $P \in F_q(x)$ of degree $n$.

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    $\begingroup$ Welcome 👋. What is $K$? And what is the definition of $X$ and $Y$? $\endgroup$ Commented Dec 25, 2024 at 3:04
  • $\begingroup$ I explained what $K$, $X$, and $Y$ were in the post. $\endgroup$ Commented Dec 25, 2024 at 18:00
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    $\begingroup$ No. There are many function fields, there is not "the function field". Also, $X,Y$ could be any linear maps as it seems, since you didn't define them, then how are we supposed to answer the question how to compute their inverses? $\endgroup$ Commented Dec 25, 2024 at 18:11
  • $\begingroup$ I edit the post with the definitions of $X$ and $Y$. $\endgroup$ Commented Dec 25, 2024 at 18:38
  • $\begingroup$ If $X$ and $Y$ are linear mappings, they can be converted into matrices, in which case the inverse matrices would correspond to $X^{-1}$ and $Y^{-1}$ $\endgroup$ Commented Dec 26, 2024 at 7:59

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