广义几何Goppa码的相等性与iso-对偶性
Equality and iso-duality of Generalized Geometric Goppa codes
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中文总结 AI 辅助
本文建立广义几何Goppa码相等的充要条件，并据此刻画iso-对偶与自对偶码，同时将构造扩展到可分扩张域上。
中文摘要 AI 辅助
几何Goppa码是通过在次数为一的位点处对代数函数域的元素求值而构造的。扩展这一概念，[XNL99]和[Hey02]中通过允许在任意次数的位点处求值，定义了广义几何Goppa码。本文遵循[MP93]的方法，建立了广义几何Goppa码相等的充分必要条件。这些条件是我们刻画iso-对偶和自对偶广义几何Goppa码的关键工具。给定代数函数域的可分扩张M/F以及F上的一个iso-对偶广义几何Goppa码，我们扩展了[CPQT24]的构造，得到M上的iso-对偶广义几何Goppa码。此外，我们研究了广义AG码（[XNL99]，[DM03]）的自对偶性。
英文摘要
Geometric Goppa codes are constructed by evaluating elements of an algebraic function field at the places of degree one. Extending this, generalized geometric Goppa codes were defined in [XNL99] and [Hey02] by allowing evaluation at places of arbitrary degree. In this paper, we establish necessary and sufficient conditions for equality of generalized geometric Goppa codes, following the approach of [MP93]. These conditions serve as a key tool in our characterization of iso-dual and self-dual generalized geometric Goppa codes. Given a separable extension M/F of algebraic function fields and an iso-dual generalized geometric Goppa code over F, we extend the construction of [CPQT24] to obtain iso- dual generalized geometric Goppa codes over M. Furthermore, we study self-duality of generalized AG codes ([XNL99], [DM03]).
