Construction of codes on arbitrary plane curves

Abstract

In 1981, Goppa established an epoch-making method for constructing a code utilizing the theory of algebraic curves [1]. This is called the algebraic curve code. This paper develops the theory on an arbitrary curve, which may contain singular points and discusses the method of constructing the code. The method employed in this paper is to eliminate the singular point by applying the transformation called blowing-up and to construct the code on the nonsingular model. The number of genes of curves can be derived simultaneously. It is made easy to derive the bases of the linear spaces L(G) and Ω(G-D), as in the case of the nonsingular curve. The algebraic curve C considered in this paper is the irreducible projected plane curve or the algebraic curve which is obtained by gluing irreducible affine plane curves. In this case, the nonsingular model of C can also be represented as a gluing of irreducible affine plane curves. Then the method is applied to the hyperelliptic plane curve on which the number of rational points achieves the Hasse-Weil bound or certain hypernonelliptic plane curves. It is shown that the precise structure of the code can be examined.