Applications of Semigroups to Algebraic Geometry Codes

研究人员研究了代数几何在编码理论中的应用。在这个方案中，研究者集中在与半群和代数几何码有关的问题上。这涉及到探索曲线上m元组点的魏尔斯特拉斯半群与使用这m个点形成的代数几何代码之间的联系。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m元组点集的魏尔斯特拉斯半群的结构。第二个目标是应用这一知识来改进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。这是建立在研究者先前的工作的基础上的，该工作表明点对的魏尔斯特拉斯半群可以用来构造参数超过通常下界的代数几何码。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，在这个项目的第三个组成部分中，研究人员将使用曲线上的m个点构造的代码与使用少于m个点构造的代码进行比较。除了这些主题，这个项目还包括对用于代数几何码的译码算法的半群的关注。纠错码用于通过检测和纠正错误来确保在有噪声的通信信道上可靠地传输信息。从计算设备到蜂窝电话再到CD播放机，这样的代码可以在广泛的设备中找到。在实际应用中，编码应该是高效的，并能纠正尽可能多的错误。虽然构造代码的方法有很多种，但最有希望的方法之一是使用代数几何中的工具。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。在这个方案中，研究者通过研究与曲线相关的半群来研究通常用于定义代数几何码的曲线。然后应用该知识来获得对相应代码的效率和纠错能力的更好估计。此外，作者还探讨了半群在代数几何码译码过程中的应用。

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.