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Date Available
4-29-2016
Year of Publication
2016
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Faculty
Dr. Heide Gluesing-Luerssen
Faculty
Dr. Peter Perry
Abstract
Cyclic codes are a well-known class of linear block codes with efficient decoding algorithms. In recent years they have been generalized to skew-constacyclic codes; such a generalization has previously been shown to be useful. We begin with a study of skew-polynomial rings so that we may examine these codes algebraically as quotient modules of non-commutative skew-polynomial rings. We introduce a skew-generalized circulant matrix to aid in examining skew-constacyclic codes, and we use it to recover a well-known result on the duals of skew-constacyclic codes from Boucher/Ulmer in 2011. We also motivate and develop a notion of idempotent elements in these quotient modules. We are particularly concerned with the existence and uniqueness of idempotents that generate a given submodule; we generalize relevant results from previous work on skew-constacyclic codes by Gao/Shen/Fu in 2013 and well-known results from the classical case.
Digital Object Identifier (DOI)
http://dx.doi.org/10.13023/ETD.2016.169
Recommended Citation
Fogarty, Neville Lyons, "On Skew-Constacyclic Codes" (2016). Theses and Dissertations--Mathematics. 36.
https://uknowledge.uky.edu/math_etds/36
