Problems in Algebraic Coding Theory

摘要：Walker dms -0302024在信道上传输信息时，错误是不可避免的。编码理论的目标是找到有效的方法对信息进行编码，从而纠正这些错误。代数编码理论试图使用诸如线性代数、群论和代数几何等领域的技术来做到这一点。本项目涉及代数编码理论中的四个主要问题：寻找特定环上代数几何码的有效解码算法，研究特定环上的指数和及其与编码理论的联系，从结构角度研究码，以及构造特征大于2的量子码。这个项目的大部分内容涉及到PI将与她的研究生和博士后学生一起完成的工作。此外，她将继续她在研究之外的数学活动中非常活跃的历史。这些活动包括为高中女生组织一个为期一周的暑期数学营，以及为女大学生数学家组织一次全国研究会议。在以前由NSF支持的时候，PI是IAS/Park City数学研究所指导项目的本科讲师，在当前项目期间，她将成为“重新连接”研讨会的首席讲师，该研讨会由主要以教学为重点的机构的教师组成。更准确地说，这个项目中包含的问题描述如下。在以前的工作中，PI在局部环上引入了代数几何码，并证明了关于这些码的几个基本结果。本项目的两个部分建立在之前的工作基础上。首先，在与R. Koetter的合作中，PI打算为这些代码找到一种有效的解码算法。该算法将对平方欧几里得和齐次权测度进行解码。该算法的设计将依赖于苏丹的列表解码算法和苏丹算法的kotter - vardy软决策。其次，PI将建立在她过去关于环上曲线的指数和及其与代码的关系的研究基础上。而她之前的工作(与j - f。Voloch)专注于Weil指数和及其在局部Artinian环上代数几何码的齐次和平方欧几里得权中的应用，其他类型的和在有限域上的码的研究中被证明是有价值的，pi现在将研究这些其他和的环类似物及其应用。这个项目的第三个领域建立在PI对编码理论的结构方法的优先工作的基础上。她之前用这种方法重新审视了二进制自对偶码的分类，除了继续这项工作，她还将从这个角度研究三元自对偶码以及加性GF(4)码。这个项目的第四个也是最后一个主要领域是由代数曲线产生的量子代数几何编码。和j - l一起。Kim和T. Marley， PI打算构造特征大于2的良好代数几何量子码。

Abstract for award of Walker DMS-0302024Whenever information is transmitted across a channel, errors are boundto occur. It is the goal of coding theory to find efficient ways toencode the information so that these errors can be corrected.Algebraic coding theory seeks to do this using techniques from areassuch as linear algebra, group theory, and algebraic geometry. Thisproject involves four main problems within algebraic coding theory:the search for an efficient decoding algorithm for algebraic geometriccodes over certain rings, the study of exponential sums over certainrings and their connections to coding theory, the study of codes froma structural standpoint, and the construction of quantum codes incharacteristics greater than 2. Much of this project involves workthe PI will do with her graduate and postdoctoral students. Inaddition, she will continue her history of being extremely active inmathematical activities outside her research. These activitiesinclude organizing a week-long summer math camp for high school girlsand a national research conference for undergraduate womenmathematicians. While previously supported by NSF, the PI was theundergraduate lecturer at the IAS/Park City Mathematics InstituteMentoring Program, and during the term of the current project, shewill be the principal lecturer at a "Reconnect" workshop for facultyfrom primarily teaching-focused institutions.More precisely, the problems included in this project are described asfollows. In previous work, the PI introduced algebraic geometriccodes over local Artinian rings and proved several foundationalresults about these codes. Two portions of this project build on thisprior work. First, in joint work with R. Koetter, the PI intends tofind an efficient decoding algorithm for these codes. This algorithmwill decode with respect to the squared Euclidean and homogeneousweight measures. The design of the algorithm will depend both uponSudan's list decoding algorithm and the Koetter-Vardy soft-decisionversion of Sudan's algorithm. Second, the PI will build upon her pastwork with exponential sums along curves over rings and theirrelationship to codes. While her previous work (joint withJ.-F. Voloch) has focused on Weil exponential sums and theirapplications to homogeneous and squared Euclidean weights of algebraicgeometric codes over local Artinian rings, other types of sums haveproven valuable in the study of codes over finite fields and the PIwill now study the ring analogues of these other sums and theirapplications. The third area of this project builds on the PI's priorwork on a structural approach to coding theory. She previously usedthis approach to revisit the classification of binary self-dual codes,and in addition to continuing this work, she will look at ternaryself-dual codes as well as additive GF(4)-codes from this point ofview. The fourth and final main area of this project is that ofquantum algebraic-geometric codes arising from algebraic curves.Together with J.-L. Kim and T. Marley, the PI intends to constructgood algebraic geometric quantum codes in characteristics greater than2.