Quasi-cyclic codes with cyclic constituent codes

Abstract

Ling and Solé [S. Ling, P. Solé, On the algebraic structure of quasi-cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47 (2001) 2751–2760] showed that every quasi-cyclic code C is constructed from shorter linear codes which are called the constituent codes of C. Given a quasi-cyclic code C of length ℓm and index ℓ with m being pairwise coprime to ℓ and the order of the field C is over, if all its constituent codes are cyclic with their zeroes having full multiplicity, C is then shown to be equivalent to a cyclic code whose zeroes with their multiplicities are fully described in terms of the nonzeroes of the cyclic constituent codes. The general transformation to obtain the above-mentioned equivalent cyclic code is also explicitly given. The approach adopted here follows the approach used by A.M.A. Natividad [A.M.A. Natividad, PhD thesis, Department of Mathematics, University of Philippines Diliman, The Philippines, 2004] and uses the generalized discrete Fourier transform on the algebraic structure of the class of quasi-cyclic codes developed by Ling and Solé [S. Ling, P. Solé, On the algebraic structure of quasi-cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47 (2001) 2751–2760].