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Haboush, William Duursma, Iwan Katz, Sheldon Dodd, Christoper Hong, Euijin 2019-08-23T19:51:42Z 2019-08-23T19:51:42Z 2019-04-17 2019-05 This thesis consists of two parts. In the first half, we define, so called, generalized Artin-Schreier cover of a scheme X over k. After defining Artin-Schreier group scheme Γ over X, a generalized Artin-Schreier cover is realized as a principal homogeneous space of Γ. We are especially interested in the case when X is P1\{0,1,∞}, a thrice punctured plane. An argument of (generalized) Artin-Schreier field extension and its function field arithmetic follows. The second half is about the coding theory. For a full flag of codes, if it is equivalent to its duals, then it is said to have the isometry-dual property. Introducing characterizations of isometry-dual property for one-point AG codes and its preservation after puncturing at some points, some generalizations in different directions will be given. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-08-22 without embargo terms The student, Euijin Hong, accepted the attached license on 2019-04-15 at 14:21. The student, Euijin Hong, submitted this Dissertation for approval on 2019-04-15 at 14:28. This Dissertation was approved for publication on 2019-04-17 at 08:05. DSpace SAF Submission Ingestion Package generated from Vireo submission #13513 on 2019-08-22 at 14:41:52 Made available in DSpace on 2019-08-23T19:51:42Z (GMT). No. of bitstreams: 2 HONG-DISSERTATION-2019.pdf: 369570 bytes, checksum: 74bdf911c837bc9b4f44c75a879a67ed (MD5) LICENSE.txt: 4208 bytes, checksum: 148dfe8063e05d9769f6cfae27c95add (MD5) Previous issue date: 2019-04-17 application/pdf http://hdl.handle.net/2142/104786 en ⃝c 2019 by Euijin Hong. All rights reserved. Algebraic Geometry Algebraic Geometry Code Two problems in the theory of curves over fields of positive characteristic text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.