Code Equivalence and Automorphism Groups of Weighted Superelliptic Codes

Source abstract

We study permutation and monomial equivalence for one-point codes on weighted superelliptic curves. On strata where $H_m\simeq \Z/n\Z\times\Z/m\Z$ acts regularly on the evaluation set, we prove that these codes are Cartesian character codes and determine their primal and dual minimum distances. We compute the projective monomial normalizer of the regular action and classify the compatible permutation and monomial equivalences through the induced actions on the defining sets. When $d > n$ and $m \geq 2$, the common permutation automorphism group of the full filtration is exactly $H_m$, and the regular action can be recovered from its two lowest nonconstant members. We determine the full permutation automorphism group in closed form at the top of the filtration and give a field-independent finite certificate for $\PAut ( \CC_s ) = H_m$ in the remaining primitive cases treated here. The hull dimension is bounded by the genus on every regular stratum, apart from the type $(3,3,3)$. Explicit examples show that length, dimension, and hull dimension do not determine monomial equivalence, and that the hull profile together with the regular automorphism subgroup does not determine the curve up to isomorphism. As an application, we determine the corresponding hull-based invariants and punctured signatures used in support-splitting methods.