Assume you have an overdetermined equation system $Ax = b$ over $Z_p$ with $p$ prime, that is $A \in \mathbb{Z}_p^{m \times n}$, $x \in \mathbb{Z}_p^{n}$, $b \in \mathbb{Z}_p^{m}$ ($x$ unknown) and the euclidean norm defined as $$\|w\| := \sqrt{\|w_0\|_{\infty}^2 + \cdots + \|w_n\|_{\infty}^2}$$ where $\|w_i\|_{\infty} = |w_i|$ with the representation chosen for $\mathbb{Z}_p$ is $\{-(p-1)/2, \dots, (p-1)/2\}$. I am looking for an $x^{*} \in \mathbb{Z}_p^n$ that minimizes $\|b - A x^{*}\|_2$.
I know that in a setting over real numbers, it holds that the Moore-Penrose Pseudoinverse $A^{\dagger}$ is a matrix such that choosing $x^{*} = A^{\dagger}b$ minimizes $\|b - A x^{*}\|_2$.
Does the concept hold for the Moore-Penrose pseudoinverse over this "custom" norm and calculations done over the field $\mathbb{Z}_p$? If so, what algorithm can be used to calculate the Moore-Penrose pseudoinverse in this setting?