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

Abstract

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.

Keywords

• Algebraic geometric codes,Asymptotically good ramp secret sharing schemes,Generalized Hamming weights,Relative generalized Hamming weights,Secret sharing

References

1. [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. [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. [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. [4] O. Geil, S. Martin, Relative generalized Hamming weights of q-ary Reed–Muller codes, to appear in Adv. Appl. Commun.
5. [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. [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. [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. [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. [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. [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. [11] M. A. Tsfasman, S. G. Vladuµ, Geometric approach to higher weights, IEEE Trans. Inform. Theory 41(6, part 1) (1995) 1564–1588.
12. [12] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory 37(5) (1991) 1412–1418.
13. [13] J. Zhang K. Feng, Relative generalized Hamming weights of cyclic codes, arXiv preprint arXiv:1505.07277, 2015.