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$F_q$-linear cyclic codes over $F_qm$: DFT approach

Dey, Bikash Kumar and Sundar Rajan, B (2005) $F_q$-linear cyclic codes over $F_qm$: DFT approach. In: Designs, Codes and Cryptography, 34 (1). pp. 89-116.

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Abstract

Codes over $F_qm$ that are closed under addition, and multiplication with elements from $F_q$ are called $F_q$ -linear codes over $F_qm$. For $m\neq1$, this class of codes is a subclass of nonlinear codes. Among $F_q$ -linear codes, we consider only cyclic codes and call them $F_q$ -linear cyclic codes ($F_qLC$ codes) over $F_qm$. The class of $F_qLC$ codes includes as special cases (i) group cyclic codes over elementary abelian groups (q = p, a prime), (ii) subspace subcodes of Reed–Solomon codes ($n = q^m-1$) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over $F_q$ (m = 1) and (iv) twisted BCH codes. Moreover, with respect to any particular $F_q$ -basis of $F_qm$, any $F_qLC$ code over $F_qm$ can be viewed as an m-quasi-cyclic code of length mn over $F_q$ . In this correspondence, we obtain transform domain characterization of $F_qLC$ codes, using Discrete Fourier Transform (DFT) over an extension field of $F_qm$. The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over $F_qm$. We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual $F_qLC$ codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Springer.
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited: 20 Jun 2007
Last Modified: 19 Sep 2010 04:38
URI: http://eprints.iisc.ernet.in/id/eprint/11207

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