Enumeration of Generalized Necklaces over Finite Fields




Algallaf, Jumah Ali

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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.



Generalized Necklaces, Finite Field, QC codes, QT codes