Whenever information is transmitted across a channel, errors are bound to occur. It is the goal of coding theory to find efficient ways to encode the information so that these errors can be corrected. Algebraic coding theory seeks to do this using techniques from areas such as linear algebra, group theory, and algebraic geometry. This project involves four main problems within algebraic coding theory: the search for an efficient decoding algorithm for algebraic geometric codes over certain rings, the study of exponential sums over certain rings and their connections to coding theory, the study of codes from a structural standpoint, and the construction of quantum codes in characteristics greater than 2. Much of this project involves work the PI will do with her graduate and postdoctoral students. In addition, she will continue her history of being extremely active in mathematical activities outside her research. These activities include organizing a week-long summer math camp for high school girls and a national research conference for undergraduate women mathematicians. While previously supported by NSF, the PI was the undergraduate lecturer at the IAS/Park City Mathematics Institute Mentoring Program, and during the term of the current project, she will be the principal lecturer at a "Reconnect" workshop for faculty from primarily teaching-focused institutions.
More precisely, the problems included in this project are described as follows. In previous work, the PI introduced algebraic geometric codes over local Artinian rings and proved several foundational results about these codes. Two portions of this project build on this prior work. First, in joint work with R. Koetter, the PI intends to find an efficient decoding algorithm for these codes. This algorithm will decode with respect to the squared Euclidean and homogeneous weight measures. The design of the algorithm will depend both upon Sudan's list decoding algorithm and the Koetter-Vardy soft-decision version of Sudan's algorithm. Second, the PI will build upon her past work with exponential sums along curves over rings and their relationship to codes. While her previous work (joint with J.-F. Voloch) has focused on Weil exponential sums and their applications to homogeneous and squared Euclidean weights of algebraic geometric codes over local Artinian rings, other types of sums have proven valuable in the study of codes over finite fields and the PI will now study the ring analogues of these other sums and their applications. The third area of this project builds on the PI's prior work on a structural approach to coding theory. She previously used this approach to revisit the classification of binary self-dual codes, and in addition to continuing this work, she will look at ternary self-dual codes as well as additive GF(4)-codes from this point of view. The fourth and final main area of this project is that of quantum algebraic-geometric codes arising from algebraic curves. Together with J.-L. Kim and T. Marley, the PI intends to construct good algebraic geometric quantum codes in characteristics greater than 2.
