On generalized Hamming weights of codes constructed on affine algebraic sets
As a generalization of conventional algebraic geometric codes, codes constructed on affine algebraic sets were proposed by S. Miura. He has also shown that if a monomial order and a Gröbner basis are given for the code on an affine algebraic set, a lower bound for the minimum distance is obtained as a generalization of Feng-Rao designed distance. In this paper, we investigate their generalized Hamming weights. We first provide a lower bound for generalized Hamming weights by using the monomial order structure of the Gröbner basis employed. Secondary, by introducing a number g*, which is also determined by the monomial order structure of the Gröbner basis, we show that when the order μ, of generalized Hamming weights is greater than g*, the proposed lower bound agrees with the true generalized Hamming weights.
Unable to display preview. Download preview PDF.
- 2.K. Y. Yang, P. V. Kumar and H. Stichtenoth, “On the Weight Hierarchy of Geometric Goppa Codes,” IEEE Trans. Inform. Theory, vol. IT-40, No.3, pp. 913–920, 1994.Google Scholar
- 3.C. Munuera, “On the Generalized Hamming Weights of Geometric Goppa Codes,” IEEE Trans. Inform. Theory, vol. IT-40, No.6, pp. 2092–2099, 1994.Google Scholar
- 4.S. Miura, “On the generalized Hamming weights of Geometric Goppa Codes,” Proc. of the 1993 IEICE Fall Conference, vol. 1, pp. 316–317, 1993.Google Scholar
- 5.S. Miura, “Geometric Goppa Codes on Affine Algebraic Variety,” Proc. of 18th SITA, pp.243–246, 1995 (in Japanese).Google Scholar
- 6.R. Pellikaan, “The shift bound for cyclic, Reed-Muller and geometric Goppa code,” preprint.Google Scholar
- 7.D. Cox, J. Little and D. O'shea, Ideals, varieties and algorithms; an introduction to computational algebraic geometry and commutative algebra, Springer, Berlin, 1992.Google Scholar