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

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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.

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Acknowledgements

The first author is supported by the Project MTM2015-65764-C3-1-P (MINECO/FEDER). The second author is supported by the Project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343. The third author is supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).

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Authors and Affiliations

  1. IMUVa and Departamento de Matemática Aplicada, Escuela de Ingeniería Informática de Segovia, Universidad de Valladolid, Valladolid, Spain

    José I. Farrán

  2. IEMath-GR and Departamento de Álgebra, Universidad de Granada, 18071, Granada, Spain

    Pedro A. García-Sánchez

  3. Departamento de Matemática e Centro de Matemática e Aplicaçoes (CMA), FCT, Universidade Nova de Lisboa, Lisbon, Portugal

    Benjamín A. Heredia

Authors
  1. José I. Farrán
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  2. Pedro A. García-Sánchez
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  3. Benjamín A. Heredia
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Correspondence to José I. Farrán.

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G. Korchmaros.

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Farrán, J.I., García-Sánchez, P.A. & Heredia, B.A. On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups. Des. Codes Cryptogr. 86, 2893–2916 (2018). https://doi.org/10.1007/s10623-018-0483-4

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  • DOI: https://doi.org/10.1007/s10623-018-0483-4

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