Topics in the Theory of Algebraic Geometric Codes

摘要： 该奖项支持代数几何码领域的初步研究。 在早期的工作中，主要研究者通过引入环上的代数几何码，将代数几何码的研究与编码理论中另一个活跃而肥沃的领域结合起来。 她证明了这些新代码的几个基本结果，从而奠定了应用的基础。 这些申请将继续进行。 特别是，PI将开始通过导出这些代码的最小欧几里德和李权重上的良好界，这将导致具有良好参数的新的非线性二进制码的构造。 她还将研究定义代数几何代码的可行性来自阿贝尔计划在环。 此外，PI将开始在编码理论的不同领域的研究，即来自有限域上的曲线的代数几何码的解码算法。

Abstract: This award supports preliminary research in the area of algebraic geometric codes. In earlier work, the principal investigator combined the study of algebraic geometric codes with another active and fertile area within coding theory by introducing algebraic geometric codes over rings. She proved several foundational results on these new codes, thus laying the groundwork for applications. These applications will now be pursued. In particular, the PI will begin by deriving a good bound on the minimum Euclidean and Lee weights of these codes, which should lead to the construction of new nonlinear binary codes with good parameters. She will also investigate the feasibility of defining algebraic geometric codes coming from abelian schemes over rings. In addition, the PI will begin research in a different area within coding theory, that of decoding algorithms for algebraic geometric codes coming from curves over finite fields.