Applications of Semigroups to Algebraic Geometry Codes

研究者研究代数几何在编码理论中的应用。在这个建议中，研究者着重于半群和代数几何码的相关问题。这涉及探索曲线上点的m元组的Weierstrass半群与使用这些m点形成的代数几何代码之间的联系。本项目的第一个目的是更好地理解有限域上某些曲线上点的m元组的Weierstrass半群的结构。第二个目标是应用这些知识来改进使用这些曲线以及有限域上任意曲线构造的代码的最小距离界限。这建立在研究者以前的工作上，表明Weierstrass点对半群可以用来构造参数超过通常下界的代数几何代码。这些代码使用曲线上的两点构造，比使用同一曲线上的单点构造的任何可比代码具有更好的参数。因此，在这个项目的第三个组成部分中，研究者比较了在曲线上使用m个点构建的代码和使用少于m个点构建的代码。除了这些主题之外，本项目还包括对代数几何码解码算法中使用的半群的关注。纠错码通过检测和纠错来保证信息在噪声通信信道中的可靠传输。从计算设备到移动电话再到激光唱机，这些代码广泛存在于各种设备中。在实际应用中，代码应该是高效的，并尽可能多地纠正错误。虽然有许多构建代码的方法，但最有前途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于定义这些代码的曲线来获得有关这些代码的信息。在这个建议中，研究者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更好地估计相应代码的效率和纠错能力。此外，研究者追求半群在解码代数几何码过程中的应用。

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.