Applications of Semigroups to Algebraic Geometry Codes

调查研究应用代数几何编码理论。 在这个建议中，调查员集中在有关半群和代数几何码的问题。 这涉及到探索魏尔斯特拉斯半群的m元组的点在曲线和代数几何代码形成使用这些m点之间的连接。 这个项目的第一个目标是更好地理解有限域上某些曲线上的点的m元组的Weierstrass半群的结构。 第二个目标是应用这些知识来提高使用这些曲线以及有限域上的任意曲线构造的代码的最小距离的界限。 这是建立在调查员以前的工作表明，维尔斯特拉斯半群的点对可以用来构建代数几何代码的参数超过通常的下限。 这些代码使用曲线上的两个点构造，具有比使用同一曲线上的单个点构造的任何可比代码更好的参数。 因此，在本项目的第三部分，研究人员比较了使用m个曲线上的点构建的代码与使用少于m个点构建的代码。 除了这些主题，这个项目还包括一个重点是半群用于解码算法的代数几何代码。 纠错码用于通过检测和纠正错误来确保信息在有噪声的通信信道上的可靠传输。 从计算机设备到蜂窝电话再到光盘播放器，在各种各样的设备中都可以找到这种代码。 对于实际应用，代码应该是有效的，并纠正尽可能多的错误。 虽然有许多方法可以构造代码，但最有前途的方法之一是使用代数几何工具。 特别地，有限域上的曲线被用来定义纠错码，称为代数几何码。 人们可以通过研究用来定义它们的曲线来获得关于这些代码的信息。 在这个建议中，研究人员通过研究与曲线相关的半群来检查通常用于定义代数几何代码的曲线。 然后应用这些知识来获得相应代码的效率和纠错能力的更好估计。 此外，调查追求的应用程序的半群解码代数几何代码的过程中。

The investigator studies applications of algebraic geometry to coding theory. In this proposal, the investigator focuses on problems relating semigroups and algebraic geometry codes. This involves exploring connections between Weierstrass semigroups of m-tuples of points on a curve and algebraic geometry codes formed using these m points. The first aim of this project is to better understand the structure of Weierstrass semigroups of m-tuples of points on certain curves over finite fields. The second goal is to apply this knowledge to improve bounds on the minimum distances of codes constructed using these curves as well as arbitrary curves over finite fields. This builds on the investigator's previous work showing that Weierstrass semigroups of pairs of points can be used to construct algebraic geometry codes with parameters exceeding the usual lower bounds. These codes, constructed using two points on a curve, have better parameters than any comparable code constructed using a single point on the same curve. Thus, in the third component of this project, the investigator compares codes constructed using m points on a curve to those constructed using fewer than m points. In addition to these topics, this project includes a focus on semigroups used in decoding algorithms for algebraic geometry codes. Error-correcting codes are used to ensure reliable transfer of information across noisy communication channels by detecting and correcting errors. Such codes are found in a wide range of devices, from computing equipment to cellular telephones to compact disc players. For practical applications, codes should be efficient and correct as many errors as possible. While there are many ways to construct codes, one of the most promising uses tools from algebraic geometry. In particular, curves over finite fields are used to define error-correcting codes, called algebraic geometry codes. One can gain information about these codes by studying the curves used to define them. In this proposal, the investigator examines curves commonly used to define algebraic geometry codes by studying semigroups associated with the curves. This knowledge is then applied to obtain better estimates of the efficiency and error-correcting capabilities of the corresponding codes. In addition, the investigator pursues applications of semigroups to the process of decoding algebraic geometry codes.