代码和Cartier操作符
Codes and the Cartier Operator
摘要：
本文提出了一种新的由代数曲线构造码的方法。给定非素有限域上的一条曲线，所得到的码定义在一个子域上。我们称它们为卡地亚码，因为它们的构造涉及卡地亚运算符。这类新的代码可以被看作是一个自然的几何推广的经典Goppa码。特别是，我们证明了一个著名的性质满足经典Goppa码自然扩展到卡地亚码。我们证明了这些码的维数和最小距离的一般下界，并将我们的构造与经典的代数几何码的子域子码进行了比较。我们证明了每一个Cartier码都包含在代数几何码的子域子码中，并且这两种码的构造具有相似的渐近性能。利用Cartier算子的性质可以很好地证明关于代数几何码的子域子码的一些已知结果，并且由于Cartier码和Cartier算子，代数几何码的子域子码的维数的一些已知界可以得到改进.
英文摘要：
In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier Codes since their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances. We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.
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