Locally Recoverable Codes From Algebraic Curves

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dc.contributor.advisorBarg, Alexanderen_US
dc.contributor.advisorHaines, Thomasen_US
dc.contributor.authorBallentine, Sean Fredericken_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2019-02-01T06:41:26Z
dc.date.available2019-02-01T06:41:26Z
dc.date.issued2018en_US
dc.description.abstractLocally recoverable (LRC) codes have the property that erased coordinates can be recovered by retrieving a small amount of the information contained in the entire codeword. An LRC code achieves this by making each coordinate a function of a small number of other coordinates. Since some algebraic constructions of LRC codes require that $n \leq q$, where $n$ is the length and $q$ is the size of the field, it is natural to ask whether we can generate codes over a small field from a code over an extension. Trace codes achieve this by taking the field trace of every coordinate of a code. In this thesis, we give necessary and sufficient conditions for when the local recoverability property is retained when taking the trace of certain LRC codes. This thesis also explores a subfamily of LRC codes with hierarchical locality (H-LRC) which have tiers of recoverability. We provide a general construction of codes with 2 levels of hierarchy from maps between algebraic curves and present several families from quotients of curves by a subgroup of automorphisms. We consider specific examples from rational, elliptic, Kummer, and Artin-Schrier curves and examples of asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.en_US
dc.identifierhttps://doi.org/10.13016/l3qx-sd21
dc.identifier.urihttp://hdl.handle.net/1903/21655
dc.language.isoenen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pqcontrolledElectrical engineeringen_US
dc.subject.pquncontrolledCodeen_US
dc.subject.pquncontrolledCoding Theoryen_US
dc.subject.pquncontrolledCurvesen_US
dc.subject.pquncontrolledLocally Recoverableen_US
dc.titleLocally Recoverable Codes From Algebraic Curvesen_US
dc.typeDissertationen_US

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