Enumeration of Generalized Necklaces over Finite Fields
Date
2016-04-28
Authors
Algallaf, Jumah Ali
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Abstract
In combinatorial theory, a necklace is an equivalence class of a word under cyclic shift. Enumerating necklaces over a finite field F_q is an essential yet time-consuming step in constructing Quasi-Cyclic (QC) and Quasi-Twisted (QT) codes. QC and QT codes are important subclasses of linear block codes which can be characterized in terms of m×m circulant and twistulant matrices, respectively. Circulant and twistulant matrices have been used extensively in the construction of error correcting codes and many of the best-known linear codes have been obtained using constructions based on theses matrices. In this project, a generalization of necklaces which is related to circulant and twistulant matrices is presented along with a closed form expression to count their numbers. The goal is to enumerate these generalized necklaces over prime and prime power fields using MATLAB.
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Keywords
Generalized Necklaces, Finite Field, QC codes, QT codes