On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

Delgado, M.; Farrán, J.; García-Sánchez, P.; Llena, D.

doi:10.1090/S0025-5718-2013-02673-7

On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

Authors: M. Delgado, J. I. Farrán, P. A. García-Sánchez and D. Llena
Journal: Math. Comp. 82 (2013), 1813-1836
MSC (2010): Primary 20M14, 11Y55, 11T71
DOI: https://doi.org/10.1090/S0025-5718-2013-02673-7
Published electronically: January 28, 2013
MathSciNet review: 3042586
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{\mathrm {th}}$ Feng-Rao number is obtained.

1. Angela I. Barbero and Carlos Munuera, The weight hierarchy of Hermitian codes, SIAM J. Discrete Math. 13 (2000), no. 1, 79–104. MR 1737936, https://doi.org/10.1137/S089548019834342X
2. A. Campillo and J. I. Farrán, Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models, Finite Fields Appl. 6 (2000), no. 1, 71–92. MR 1738217, https://doi.org/10.1006/ffta.1999.0266
3. Antonio Campillo, José Ignacio Farrán, and Carlos Munuera, On the parameters of algebraic-geometry codes related to Arf semigroups, IEEE Trans. Inform. Theory 46 (2000), no. 7, 2634–2638. MR 1806823, https://doi.org/10.1109/18.887872
4. S. T. Chapman, P. A. García-Sánchez, and D. Llena, The catenary and tame degree of numerical monoids, Forum Math. 21 (2009), no. 1, 117–129. MR 2494887, https://doi.org/10.1515/FORUM.2009.006
5. M. Delgado, P. A. García-Sánchez and J. Morais, ``NumericalSgps'', A GAP package for numerical semigroups, current version number 0.97 (2011). Available via http://www.gap-system.org/.
6. J. I. Farrán, P. A. García-Sánchez and D. Llena, ``On the Feng-Rao numbers'', Actas de las VII Jornadas de Matemática Discreta y Algorítmica, pp. 321-333 (2010).
7. J. I. Farrán and C. Munuera, Goppa-like bounds for the generalized Feng-Rao distances, Discrete Appl. Math. 128 (2003), no. 1, 145–156. International Workshop on Coding and Cryptography (WCC 2001) (Paris). MR 1991422, https://doi.org/10.1016/S0166-218X(02)00441-9
8. Gui Liang Feng and T. R. N. Rao, Decoding algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inform. Theory 39 (1993), no. 1, 37–45. MR 1211489, https://doi.org/10.1109/18.179340
9. Petra Heijnen and Ruud Pellikaan, Generalized Hamming weights of 𝑞-ary Reed-Muller codes, IEEE Trans. Inform. Theory 44 (1998), no. 1, 181–196. MR 1486657, https://doi.org/10.1109/18.651015
10. Tor Helleseth, Torleiv Kløve, and Johannes Mykkeltveit, The weight distribution of irreducible cyclic codes with block length 𝑛₁((𝑞^{𝑙}-1)/𝑁), Discrete Math. 18 (1977), no. 2, 179–211. MR 446717, https://doi.org/10.1016/0012-365X(77)90078-4
11. Tom Høholdt, Jacobus H. van Lint, and Ruud Pellikaan, Algebraic geometry codes, Handbook of coding theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 871–961. MR 1667946
12. Christoph Kirfel and Ruud Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory 41 (1995), no. 6, 1720–1732. Special issue on algebraic geometry codes. MR 1391031, https://doi.org/10.1109/18.476245
13. J. C. Rosales and P. A. García-Sánchez, ``Numerical Semigroups'', Developments in Maths. vol. 20, Springer (2010).
14. Victor K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37 (1991), no. 5, 1412–1418. MR 1136673, https://doi.org/10.1109/18.133259

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 20M14, 11Y55, 11T71

Retrieve articles in all journals with MSC (2010): 20M14, 11Y55, 11T71

Additional Information

M. Delgado
Affiliation: CMUP, Departamento de Matematica, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: mdelgado@fc.up.pt

J. I. Farrán
Affiliation: Departamento de Matemática Aplicada, Escuela Universitaria de Informática, Campus de Segovia - Universidad de Valladolid, Plaza de Santa Eulalia 9 y 11 - 40005 Segovia, Spain
Email: jifarran@eii.uva.es

P. A. García-Sánchez
Affiliation: Departamento de Álgebra, Universidad de Granada, 18071 Granada, España
Email: pedro@ugr.es

D. Llena
Affiliation: Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, 04120 Almería, España
Email: dllena@ual.es

DOI: https://doi.org/10.1090/S0025-5718-2013-02673-7
Keywords: AG codes, weight hierarchy, numerical semigroups, order bounds, Goppa-like bounds, Feng-Rao numbers.
Received by editor(s): May 19, 2011
Received by editor(s) in revised form: November 22, 2011
Published electronically: January 28, 2013
Additional Notes: The first author was partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.
The second author was supported by the project MICINN-MTM-2007-64704.
The third and fourth authors were supported by the projects MTM2010-15595, FQM-343 and FEDER funds
The third author was also supported by the project FQM-5849.
Article copyright: © Copyright 2013 American Mathematical Society