The optimality of Feng-Rao designed minimum distance for Hermitian codes constructed by weight order based on pole order

Abstract

The algebraic geometric code is known as a linear code that guarantees a relatively large minimum distance under the condition that the number of check symbols is kept constant, when the code length is long. Recently, Saints and Heegard presented a unified theory for decoding of the algebraic geometric code and the multidimensional cyclic code, based on the monomial order and the theory of Gröbner bases. Miura, on the other hand, extended the definition of the Feng–Rao designed distance to the case of arbitrary linear code and showed, based on the affine algebraic variety and the monomial order, that the Feng–Rao designed distance is relatively large in the algebraic geometric code. In that case, the Feng–Rao designed distance of the code depends on the definition of the monomial order. It is then important, from the viewpoint of code construction, to determine the class of the monomial orders that provides the maximum Feng–Rao designed distance. This paper considers the “Hermitian code” constructed from the Hermitian curve, which is a typical class of the algebraic geometric codes, and derives the class of the monomial orders that provides the maximum Feng–Rao designed distance. It is also shown that the Feng–Rao designed distance derived from the weight order based on the pole order has the optimal property. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(11): 85–95, 2000