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Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower

Yıl 2017, , 37 - 47, 11.01.2017
https://doi.org/10.13069/jacodesmath.34390

Öz

Asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point
algebraic geometric codes defined from asymptotically good towers of function fields. Their security
is given by the relative generalized Hamming weights of the corresponding codes. In this paper we
demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is
small. For general codimension, we give an improved estimate for the highest RGHW.

Kaynakça

  • [1] M. Bras–Amorós, M. O’Sullivan, The order bound on the minimum distance of the one–point codes associated to the Garcia–Stichtenoth tower, IEEE Trans. Inform. Theory 53(11) (2007) 4241–4245.
  • [2] H. Chen, R. Cramer, S. Goldwasser, R. de Haan, V. Vaikuntanathan, Secure computation from random error correcting codes, In Advances in cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci. 4515 (2007) 291–310.
  • [3] A. Garcia, H. Stichtenoth, On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 61(2) (1996) 248–273.
  • [4] O. Geil, S. Martin, Relative generalized Hamming weights of q-ary Reed–Muller codes, to appear in Adv. Appl. Commun.
  • [5] O. Geil, S. Martin, U. Martínez-Peñas, R. Matsumoto, D. Ruano, Asymptotically good ramp secret sharing schemes, arXiv preprint arXiv:1502.05507, 2015.
  • [6] O. Geil, S. Martin, R. Matsumoto, D. Ruano, Y. Luo, Relative generalized Hamming weights of one–point algebraic geometric codes, IEEE Trans. Inform. Theory 60(10) (2014) 5938–5949.
  • [7] J. Kurihara, T. Uyematsu, R. Matsumoto, Secret sharing schemes based on linear codes can be precisely characterized by the relative generalized Hamming weight, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E95–A(11) (2012) 2067–2075.
  • [8] Z. Liu, W. Chen, Y. Luo, The relative generalized Hamming weight of linear q-ary codes and their subcodes, Des. Codes Cryptogr. 48(2) (2008) 111–123.
  • [9] Y. Luo, C. Mitrpant, A. J. Han Vinck, K. Chen, Some new characters on the wire–tap channel of type II, IEEE Trans. Inform. Theory 51(3) (2005) 1222–1229.
  • [10] R. Pellikaan, H. Stichtenoth, F. Torres, Weierstrass semigroups in an asymptotically good tower of function fields, Finite Fields Appl. 4(4) (1998) 381–392.
  • [11] M. A. Tsfasman, S. G. Vladuµ, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41(6, part 1) (1995) 1564–1588.
  • [12] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37(5) (1991) 1412–1418.
  • [13] J. Zhang K. Feng, Relative generalized Hamming weights of cyclic codes, arXiv preprint arXiv:1505.07277, 2015.
Yıl 2017, , 37 - 47, 11.01.2017
https://doi.org/10.13069/jacodesmath.34390

Öz

Kaynakça

  • [1] M. Bras–Amorós, M. O’Sullivan, The order bound on the minimum distance of the one–point codes associated to the Garcia–Stichtenoth tower, IEEE Trans. Inform. Theory 53(11) (2007) 4241–4245.
  • [2] H. Chen, R. Cramer, S. Goldwasser, R. de Haan, V. Vaikuntanathan, Secure computation from random error correcting codes, In Advances in cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci. 4515 (2007) 291–310.
  • [3] A. Garcia, H. Stichtenoth, On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 61(2) (1996) 248–273.
  • [4] O. Geil, S. Martin, Relative generalized Hamming weights of q-ary Reed–Muller codes, to appear in Adv. Appl. Commun.
  • [5] O. Geil, S. Martin, U. Martínez-Peñas, R. Matsumoto, D. Ruano, Asymptotically good ramp secret sharing schemes, arXiv preprint arXiv:1502.05507, 2015.
  • [6] O. Geil, S. Martin, R. Matsumoto, D. Ruano, Y. Luo, Relative generalized Hamming weights of one–point algebraic geometric codes, IEEE Trans. Inform. Theory 60(10) (2014) 5938–5949.
  • [7] J. Kurihara, T. Uyematsu, R. Matsumoto, Secret sharing schemes based on linear codes can be precisely characterized by the relative generalized Hamming weight, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E95–A(11) (2012) 2067–2075.
  • [8] Z. Liu, W. Chen, Y. Luo, The relative generalized Hamming weight of linear q-ary codes and their subcodes, Des. Codes Cryptogr. 48(2) (2008) 111–123.
  • [9] Y. Luo, C. Mitrpant, A. J. Han Vinck, K. Chen, Some new characters on the wire–tap channel of type II, IEEE Trans. Inform. Theory 51(3) (2005) 1222–1229.
  • [10] R. Pellikaan, H. Stichtenoth, F. Torres, Weierstrass semigroups in an asymptotically good tower of function fields, Finite Fields Appl. 4(4) (1998) 381–392.
  • [11] M. A. Tsfasman, S. G. Vladuµ, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41(6, part 1) (1995) 1564–1588.
  • [12] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37(5) (1991) 1412–1418.
  • [13] J. Zhang K. Feng, Relative generalized Hamming weights of cyclic codes, arXiv preprint arXiv:1505.07277, 2015.
Toplam 13 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Olav Geil Bu kişi benim

Stefano Martin Bu kişi benim

Umberto Martínez-peñas Bu kişi benim

Diego Ruano Bu kişi benim

Yayımlanma Tarihi 11 Ocak 2017
Yayımlandığı Sayı Yıl 2017

Kaynak Göster

APA Geil, O., Martin, S., Martínez-peñas, U., Ruano, D. (2017). Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(1), 37-47. https://doi.org/10.13069/jacodesmath.34390
AMA Geil O, Martin S, Martínez-peñas U, Ruano D. Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. Journal of Algebra Combinatorics Discrete Structures and Applications. Ocak 2017;4(1):37-47. doi:10.13069/jacodesmath.34390
Chicago Geil, Olav, Stefano Martin, Umberto Martínez-peñas, ve Diego Ruano. “Refined Analysis of RGHWs of Code Pairs Coming from Garcia-Stichtenoth’s Second Tower”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, sy. 1 (Ocak 2017): 37-47. https://doi.org/10.13069/jacodesmath.34390.
EndNote Geil O, Martin S, Martínez-peñas U, Ruano D (01 Ocak 2017) Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. Journal of Algebra Combinatorics Discrete Structures and Applications 4 1 37–47.
IEEE O. Geil, S. Martin, U. Martínez-peñas, ve D. Ruano, “Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, ss. 37–47, 2017, doi: 10.13069/jacodesmath.34390.
ISNAD Geil, Olav vd. “Refined Analysis of RGHWs of Code Pairs Coming from Garcia-Stichtenoth’s Second Tower”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/1 (Ocak 2017), 37-47. https://doi.org/10.13069/jacodesmath.34390.
JAMA Geil O, Martin S, Martínez-peñas U, Ruano D. Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:37–47.
MLA Geil, Olav vd. “Refined Analysis of RGHWs of Code Pairs Coming from Garcia-Stichtenoth’s Second Tower”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, 2017, ss. 37-47, doi:10.13069/jacodesmath.34390.
Vancouver Geil O, Martin S, Martínez-peñas U, Ruano D. Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(1):37-4.