nach oben

Designs, Codes and Cryptography

Erschienen in:

25.10.2017

The second Feng–Rao number for codes coming from telescopic semigroups

verfasst von: José I. Farrán, Pedro A. García-Sánchez, Benjamín A. Heredia, Micah J. Leamer

Erschienen in: Designs, Codes and Cryptography | Ausgabe 8/2018

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Generalized Gabidulin codes over fields of any characteristic

Nächster Artikel On the p-ary (cubic) bent and plateaued (vectorial) functions

Apéry R.: Sur les branches superlinéaires des courbes algébriques. C. R. Acad. Sci. Paris 222, 1198–1200 (1946).MathSciNetMATH

Assi A., García-Sánchez P.A.: Constructing the set of complete intersection numerical semigroups with a given Frobenius number. Appl. Algebra Eng. Commun. Comput. 24, 133–148 (2013).MathSciNetCrossRefMATH

Assi A., García-Sánchez P.A.: Numerical Semigroups and Applicactions. RSME Springer SeriesSpringer, New York (2016).CrossRefMATH

Assi A., García-Sánchez P.A., Ojeda I.: Frobenius vectors, Hilbert series and Gluings of Affine semigroups. J. Commutative Algebra 7, 317–335 (2015).MathSciNetCrossRefMATH

Bertin J., Carbonne P.: Semi-groupes d’entiers et application aux branches. J. Algebra 49, 81–95 (1977).MathSciNetCrossRefMATH

Castellanos A.S.: Generalized Hamming weights of codes over the \(\cal{GH}\) curve. Adv. Math. Commun. 11(1), 115–122 (2017).MathSciNetCrossRefMATH

Ciolan A., García-Sánchez P.A., Moree P.: Cyclotomic numerical semigroups. Max-Plank Institute for Mathematics report 2014-64, also arXiv:1409.5614.

Delgado M., García-Sánchez P.A., Morais J.: “NumericalSgps”. A package for numerical semigroups, Version 1.0.1 (2015), (Refereed GAP package). http://www.gap-system.org.

Delgado M., Farrán J.I., García-Sánchez P.A., Llena D.: On the generalized Feng-Rao numbers of numerical semigroups generated by intervals. Math. Comput. 82, 1813–1836 (2013).MathSciNetCrossRefMATH

10.

Delgado M., Farrán J.I., García-Sánchez P.A., Llena D.: On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators. IEEE Trans. Inf. Theory 60–1, 282–295 (2014).MathSciNetCrossRefMATH

11.

Delorme C.: Sous-monoïdes d’intersection complète de \({\mathbb{N}}\). Ann. Scient. École Norm. Sup. 4(9), 145–154 (1976).CrossRefMATH

12.

Farrán J.I., García-Sánchez P.A.: The second Feng-Rao number for codes coming from inductive semigroups. IEEE Trans. Inf. Theory 61, 4938–4947 (2015).MathSciNetCrossRefMATH

13.

Farrán J.I., García-Sánchez P.A., Heredia B.A.: On the second Feng–Rao distance of Algebraic Geometry codes related to Arf semigroups. arXiv:1702.08225.

14.

Farrán J.I., Munuera C.: Goppa-like bounds for the generalized Feng-Rao distances, international workshop on coding and cryptography (WCC 2001) (Paris). Discret. Appl. Math. 128(1), 145–156 (2003).CrossRefMATH

15.

Feng G.L., Rao T.R.N.: Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39, 37–45 (1993).MathSciNetCrossRefMATH

16.

García-Sánchez P.A., Leamer M.J.: Huneke-Wiegand Conjecture for complete intersection numerical semigroup rings. J. Algebra 391, 114–124 (2013).MathSciNetCrossRefMATH

17.

Heijnen P., Pellikaan R.: Generalized Hamming weights of \(q\)-ary Reed-Muller codes. IEEE Trans. Inf. Theory 44, 181–197 (1998).MathSciNetCrossRefMATH

18.

Helleseth T., Kløve T., Mykkleveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_{1}((q^{l}-1)/N)\). Discret. Math. 18, 179–211 (1977).CrossRefMATH

19.

Høholdt T., van Lint J.H., Pellikaan R.: Algebraic Geometry codes. In: Pless V., Huffman W.C., Brualdi R.A. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).

20.

Kirfel C., Pellikaan R.: The minimum distance of codes in an array coming from telescopic semigroups. IEEE Trans. Inf. Theory 41, 1720–1732 (1995).MathSciNetCrossRefMATH

21.

Matthews G.: Codes from the Suzuki function field. IEEE Trans. Inf. Theory 50, 3298–3302 (2004).MathSciNetCrossRefMATH

22.

Matthews G., Robinson R.: A variant of the Frobenius problem and generalized Suzuki semigroups. Integers: Electron. J. Combin. Number Theory 7, A26 (2007).MATH

23.

Munuera C., Sepúlveda A., Torres F.: Generalized Hermitian codes. Des. Codes Cryptogr. 69, 123–130 (2013).MathSciNetCrossRefMATH

24.

Rosales J.C.: On presentations of subsemigroups of \({\mathbb{N}}^{n}\). Semigroup Forum 55(2), 152–159 (1997).MathSciNetCrossRefMATH

25.

Rosales J.C., García-Sánchez P.A.: Numerical Semigroups, Developments in Mathematics, vol. 20. Springer, New York (2009).CrossRefMATH

27.

The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.7.5, 2014, http://www.gap-system.org.

28.

Wei V.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37, 1412–1428 (1991).MathSciNetCrossRefMATH

29.

Zariski O.: Le problème des modules pour les courbes planes. Hermann, Paris (1986)

Titel: The second Feng–Rao number for codes coming from telescopic semigroups
verfasst von: José I. Farrán
Pedro A. García-Sánchez
Benjamín A. Heredia
Micah J. Leamer
Publikationsdatum: 25.10.2017
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 8/2018
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-017-0426-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 8/2018

On the p-ary (cubic) bent and plateaued (vectorial) functions

New upper bounds for parent-identifying codes and traceability codes

Chosen ciphertext secure keyed-homomorphic public-key cryptosystems

A construction for optimal c-splitting authentication and secrecy codes

Permutation polynomials of the type over

On the (in)efficiency of non-interactive secure multiparty computation