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Title: On the algebraic structure of quasi-cyclic codes I: Finite fields
Authors: Ling, S. 
Solé, P.
Keywords: (a + x|b + x|a + b + x) construction
(u + v|u - v) construction
(u|u + v) construction
Chinese remainder theorem (CRT)
Discrete Fourier transform (DFT)
Quasi-cyclic codes
Self-dual codes
Issue Date: Nov-2001
Citation: Ling, S., Solé, P. (2001-11). On the algebraic structure of quasi-cyclic codes I: Finite fields. IEEE Transactions on Information Theory 47 (7) : 2751-2760. ScholarBank@NUS Repository.
Abstract: A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese Remainder Theorem (CRT), or of the Discrete Fourier Transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the recent (u + v|u - v) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.
Source Title: IEEE Transactions on Information Theory
ISSN: 00189448
DOI: 10.1109/18.959257
Appears in Collections:Staff Publications

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