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On Skew-Constacyclic Codes

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.

Identiferoai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:math_etds-1035
Date01 January 2016
CreatorsFogarty, Neville Lyons
PublisherUKnowledge
Source SetsUniversity of Kentucky
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceTheses and Dissertations--Mathematics

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