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On the algebraic structure of quasi-cyclic codes .I. Finite fields

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2 Author(s)
San Ling ; Dept. of Math., Nat. Univ. of Singapore, Singapore ; P. Sole

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 (u+υ|u-υ) 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

Published in:

IEEE Transactions on Information Theory  (Volume:47 ,  Issue: 7 )