Abstract
In this manuscript we show that the second Feng–Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Apéry sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound.
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Acknowledgements
The authors would like to thank Vítor Hugo Fernandes for the helpful discussions during the preparation of this paper, and the anonymous referee for their comments. 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). The fourth author would like to thank Marco D’Anna and the rest of orginizers of the INdAM meeting: International meeting on numerical semigroups—Cortona 2014.
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Communicated by G. Korchmaros.
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Farrán, J.I., García-Sánchez, P.A., Heredia, B.A. et al. The second Feng–Rao number for codes coming from telescopic semigroups. Des. Codes Cryptogr. 86, 1849–1864 (2018). https://doi.org/10.1007/s10623-017-0426-5
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DOI: https://doi.org/10.1007/s10623-017-0426-5
Keywords
- Free Numerical semigroups
- Telescopic numerical semigroups
- Apéry sets
- Feng–Rao distances
- AG codes
- Generalized Hamming weights