Problems in Algebraic Coding Theory

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Abstract for award of Walker DMS-0302024Whenever information is transmitted across a channel, errors are boundto occur. It is the goal of coding theory to find efficient ways toencode the information so that these errors can be corrected.Algebraic coding theory seeks to do this using techniques from areassuch as linear algebra, group theory, and algebraic geometry. Thisproject involves four main problems within algebraic coding theory:the search for an efficient decoding algorithm for algebraic geometriccodes over certain rings, the study of exponential sums over certainrings and their connections to coding theory, the study of codes froma structural standpoint, and the construction of quantum codes incharacteristics greater than 2. Much of this project involves workthe PI will do with her graduate and postdoctoral students. Inaddition, she will continue her history of being extremely active inmathematical activities outside her research. These activitiesinclude organizing a week-long summer math camp for high school girlsand a national research conference for undergraduate womenmathematicians. While previously supported by NSF, the PI was theundergraduate lecturer at the IAS/Park City Mathematics InstituteMentoring Program, and during the term of the current project, shewill be the principal lecturer at a "Reconnect" workshop for facultyfrom primarily teaching-focused institutions.More precisely, the problems included in this project are described asfollows. In previous work, the PI introduced algebraic geometriccodes over local Artinian rings and proved several foundationalresults about these codes. Two portions of this project build on thisprior work. First, in joint work with R. Koetter, the PI intends tofind an efficient decoding algorithm for these codes. This algorithmwill decode with respect to the squared Euclidean and homogeneousweight measures. The design of the algorithm will depend both uponSudan's list decoding algorithm and the Koetter-Vardy soft-decisionversion of Sudan's algorithm. Second, the PI will build upon her pastwork with exponential sums along curves over rings and theirrelationship to codes. While her previous work (joint withJ.-F. Voloch) has focused on Weil exponential sums and theirapplications to homogeneous and squared Euclidean weights of algebraicgeometric codes over local Artinian rings, other types of sums haveproven valuable in the study of codes over finite fields and the PIwill now study the ring analogues of these other sums and theirapplications. The third area of this project builds on the PI's priorwork on a structural approach to coding theory. She previously usedthis approach to revisit the classification of binary self-dual codes,and in addition to continuing this work, she will look at ternaryself-dual codes as well as additive GF(4)-codes from this point ofview. The fourth and final main area of this project is that ofquantum algebraic-geometric codes arising from algebraic curves.Together with J.-L. Kim and T. Marley, the PI intends to constructgood algebraic geometric quantum codes in characteristics greater than2.