Applications of Semigroups to Algebraic Geometry Codes

Clemson University, Clemson SC

Investigators

Abstract

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.