关于允许一个完备的近权集、赋值码和Goppa码的代数
On algebras admitting a complete set of near weights, evaluation codes and Goppa codes
摘要：
1998年，Hoholdt，van Lint和Pellikaan引入了定义在F_q-代数上的“权函数”的概念，并用它构造线性码，得到了其中的一点支撑的代数几何（AG）码。后来Matsumoto证明，所有使用权重函数产生的代码实际上都是在一点上支持的AG代码。最近，“近权函数”（权函数的推广），也定义在F_q-代数上，被引入研究两点支持码。本文证明了一个代数容许一个具有相容性质的m个近权函数的集合，即该集合是一个"完备集“，当且仅当它是定义在F_q上的一条仿射几何不可约代数曲线的正则函数的环，该曲线在无穷远点上共有m个有理分支。然后，使用近权函数产生的代码正是m点上支持的AG码。给出了这些码的最小距离的计算公式，并举例说明，在某些情况下，它比通常的Goppa界更好。
英文摘要：
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.
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