On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

José I. Farrán, Pedro A. García-Sánchez, Benjamín A. Heredia

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Apéry sets, and thus several results concerning Apéry sets of Arf semigroups are presented.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalDesigns, Codes, and Cryptography
Volume86
Issue number12
DOIs
Publication statusPublished - Dec 2018

Keywords

  • AG codes
  • Arf semigroups
  • Generalized Hamming weights
  • Inductive semigroups
  • Order bounds
  • Towers of function fields

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