Miura’s Generalization of One-Point AG Codes Is Equivalent to Høholdt, van Lint and Pellikaan’s Gener- alization

Høholdt, van Lint and Pellikaan proposed a generalization of one-point AG codes called the evaluation codes, and enabled us to construct linear codes from arbitrary algebraic varieties [5], [6]. To construct an evaluation code, we find an appropriate K-algebra R with an order function w and an epimorphism φ : R → Kn of K-algebras, where K is a finite field and n is the code length. Then we define the evaluation code as a dual code of the image of some linear space of R under φ . When the order function w is a weight function, we have a lower bound for the minimum distance of an evaluation code that is a generalization of the Goppa designed minimum distance for algebraic geometry codes. Thus evaluation codes from a weight function are particularly useful, and it is interesting whether we can construct a good evaluation code from a weight function that is never obtained as an algebraic geometry code in terms of information rate and error-correcting capability. In this paper we show that if a K-algebra R has a weight function then R is the affine coordinate ring of an affine algebraic curve with exactly one place at infinity, and the epimorphism φ : R→Kn is the evaluation of elements in R at Krational points in the curve, hence the evaluation code from a weight function can be constructed as a linear code on an arbitrary algebraic curve that is proposed by Miura [9], [10]. In Miura’s generalization of one-point AG codes, the author showed that we cannot construct a better linear code from a singular curve than the traditional one-point AG code on its normalization in terms of information rate and the Feng-Rao designed minimum distance [8], hence we can construct a one-point AG code as good as a given evaluation code from a weight function in terms of information rate and the Feng-