Code Equivalence and Automorphism Groups of Weighted Superelliptic Codes
Abstract
We study permutation and monomial equivalence for one-point codes on weighted superelliptic curves. On strata where Hm≃ Z/nZ×Z/mZ 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 ≥ 2, the common permutation automorphism group of the full filtration is exactly Hm, 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 ( CCs ) = Hm 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.
Related
- Hulls, linear equivalence, and weighted superelliptic codes
- Permutation automorphism groups of cyclic codes I: cyclotomic association schemes
- The permutation and automorphism groups of univariate multiplicity codes
- Bounding Selmer Groups of Superelliptic Jacobians via Class Groups
- On the lengths of MDS codes with a two-transitive permutation automorphism group
BibTeX
@misc{beshaj2026code,
title = {{Code Equivalence and Automorphism Groups of Weighted Superelliptic Codes}},
author = {Lubjana Beshaj and Tony Shaska},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04856},
eprint = {2610.04856},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04856},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Code Equivalence and Automorphism Groups of Weighted Superelliptic Codes
AU - Lubjana Beshaj
AU - Tony Shaska
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04856
UR - https://arxiv.org/abs/2610.04856
AB - We study permutation and monomial equivalence for one-point codes on weighted superelliptic curves. On strata where H_m≃ Z/nZ×Z/mZ 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 ≥ 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.
ER -