由分离多项式定义的曲线上AG码的置换译码
Permutation Decoding of AG Codes from Curves Defined by Separated Polynomials
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中文总结 AI 辅助
本研究针对分离多项式定义曲线上的AG码，利用曲线自同构构造置换译码集，针对SAP曲线等子类实现更有效的突发错误校正，拓展了AG码的译码方法。
中文摘要 AI 辅助
本工作研究由分离多项式定义的代数曲线上产生的代数几何（AG）码的置换译码。利用基础曲线的自同构，我们构造了关联代数几何码的置换自同构，并利用所得轨道结构确定信息位与校验位。我们引入一类称为SAP曲线（分离加性多项式曲线）的曲线，研究其上定义的单点AG码。对于这些码，我们得到可校正突发错误的置换译码集，该突发错误由具有公共坐标的有理点关联坐标支撑。我们进一步识别特殊SAP曲线的子类，包括埃尔米特曲线、广义埃尔米特曲线及某些极大曲线，对于这些曲线，额外的自同构可产生更强大的译码集。
英文摘要
In this work, we investigate permutation decoding for algebraic geometry (AG) codes arising from algebraic curves defined by separated polynomials. Using automorphisms of the underlying curves, we construct permutation automorphisms of the associated algebraic geometry codes and exploit the resulting orbit structure to determine information and check positions. We introduce a class of curves, called SAP curves (Separated Additive Polynomial curves), and investigate one-point AG codes defined on them. For these codes, we obtain permutation decoding sets that correct burst errors supported on coordinates associated with rational points sharing a common coordinate. We further identify a subclass of special SAP curves, including Hermitian curves, generalized Hermitian curves, and certain maximal curves, for which additional automorphisms yield more powerful decoding sets.
