On algebras admitting a complete set of near weights, evaluation codes and Goppa codes

TL;DR

This paper characterizes algebras with complete sets of near weights as rings of regular functions on algebraic curves, linking these to algebraic-geometric codes supported on multiple points and providing a new minimum distance formula.

Contribution

It establishes a precise correspondence between algebras with complete near weight sets and algebraic curves, extending the understanding of AG codes supported on multiple points.

Findings

Algebras with complete near weight sets are rings of regular functions on algebraic curves.

Codes from near weight functions are exactly AG codes on multiple points.

A new minimum distance formula can outperform the Goppa bound in some cases.

Abstract

In 1998 Hoholdt, van Lint and Pellikaan introduced the concept of a ``weight function'' defined on a F_q-algebra and used it to construct linear codes, obtaining among them the algebraic-geometric (AG) codes supported on one point. Later it was proved by Matsumoto that all codes produced using a weight function are actually AG codes supported on one point. Recently, ``near weight functions'' (a generalization of weight functions), also defined on a F_q-algebra, were introduced to study codes supported on two points. In this paper we show that an algebra admits a set of m near weight functions having a compatibility property, namely, the set is a ``complete set'', if and only if it is the ring of regular functions of an affine geometrically irreducible algebraic curve defined over F_q whose points at infinity have a total of m rational branches. Then the codes produced using the near weight functions are exactly the AG codes supported on m points. A formula for the minimum distance of these codes is presented with examples which show that in some situations it compares better than the usual Goppa bound.