On calculation of error-correcting pairs for PELP algorithm for algebraic-geometry codes

This work explores the construction of $2$-power error locating pairs ($2$-PELP) in the context of power decoding for algebraic-geometric codes. Such a pair, consisting of codes related through the component-wise Schur product, enables unique decoding when the error weight exceeds half of the code's designed distance. So for the algebraic-geometric code $\mathcal{C}_{\mathcal{L}}(D,G)$ of the length $n$ associated with a functional field $F/\mathbb{F}_q$ of genus $g$ the ($2$-PELP) with number of errors $t \leq 2n+2g-2\deg(F)-3\deg(G)-2$ is $(\mathcal{C}_\mathcal{L}(D,F), \mathcal{C}_\mathcal{L}(D,G+F)^\bot)$, and with number of errors $t \leq 2\deg(F)-3\deg(G)+2-2g$ is $(\mathcal{C}_\mathcal{L}(D,F)^\bot,\mathcal{C}_\mathcal{L}(D,F-G))$. For the dual code $\mathcal{C}_{\mathcal{L}}(D,G)^\bot$, the ($2$-PELP) with number of errors $t \leq 3\deg(G)-2\deg(F)+4-4g-n$ is $(\mathcal{C}_\mathcal{L}(D,F),\mathcal{C}_\mathcal{L}(D,G-F))$. Furthermore, the constraints on the code divisors have been refined, and new conditions ensuring the existence of such pairs have been established.