On generalized Hamming weights of codes constructed on affine algebraic sets

Abstract

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.