A study of the mth relative Feng-Rao distances with a special emphasis on C_ab curves

@misc{campsmoreno2026study,
  title = {{A study of the $m$th relative Feng-Rao distances with a special emphasis on $C_ab$ curves}},
  author = {Eduardo Camps-Moreno and Olav Geil and Hiram H. López and Gretchen L. Matthews and Rodrigo San-José},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26649},
  eprint = {2610.26649},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26649},
}

TY  - UNPB
TI  - A study of the mth relative Feng-Rao distances with a special emphasis on C_ab curves
AU  - Eduardo Camps-Moreno
AU  - Olav Geil
AU  - Hiram H. López
AU  - Gretchen L. Matthews
AU  - Rodrigo San-José
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26649
UR  - https://arxiv.org/abs/2610.26649
AB  - In the seminal paper [13], Farrán and Munuera introduced a method to establish generic information on the mth Feng-Rao distance via a function on arbitrary numerical semigroups that serves as a lower bound on the generalized Hamming weights of dual one-point algebraic geometry codes whenever the semigroup is Weierstrass. In the present paper, we study what we shall call the primary and dual, respectively, mth relative Feng-Rao distances, which serve as lower bounds on relative generalized Hamming weights of nested primary and dual, respectively, one-point codes under similar condition. A unified method for establishing generic formulas for mth relative Feng-Rao distances of general numerical semigroups is developed and for a large variety of numerical semigroups generated by two elements, closed formula expressions are established complementing Delgado et al.s work [8] on the mth Feng-Rao distances of such semigroups. Finally, for non-singular C_ab curves, we develop a method to estimate the mth relative generalized Hamming weight of dual and primary one-point codes in those regions where the generic methods may fail to provide best possible results.
ER  -