Cyclic codes over \(R_{k}\)
type: Item
instance of: scholarly article
title: Cyclic codes over \(R_{k}\)
zbMATH Open document ID: 1237.94134
author: Steven T. Dougherty
author: Suat Karadeniz
author: Bahattin Yildiz
published in: Designs, Codes and Cryptography
publication date: 2012-03-01T00:00:00Z
review text: Afer a preliminary introduction, this paper is organized as follows: In Section 2, the authors give necessary background on linear and cyclic codes over the rings \(R_k\). In Section 3, the authors describe cyclic codes over \(R_k\) using the decompositions of the polynomial \(x^n-1\) over the ring \(R_k\). The ideals of the ring \(R_k[x]/(x^n-1)\) are described and a formula for the number of cyclic codes over \(R_k\) of certain lengths is given. Also, necessary and sufficient conditions on \(n\) are given for \(R_k[x]/(x^n-1)\) to be a local ring. In Section 4, one-generator cyclic codes over the ring \(R_k\) are classified. In particular, the authors obtain necessary and sufficient conditions on the coefficients of a polynomial in \(R_k[x]/(x^n-1)\) to generate a non-trivial cyclic code. These ideas are later used to construct some specific one-generator cyclic codes over \(R_k\) whose binary images are optimal or near-optimal. The images of cyclic codes over \(R_k\) are described in Section 5. At first, the authors consider the \(R_{k-1}\) images of odd-length cyclic codes over \(R_k\), and they prove that these are equivalent under Nechaev permutation to cyclic codes. Later, the authors prove that the binary images of cyclic codes over \(R_k\) under the Gray map \(\psi_k\) are binary quasi-cyclic codes of degree \(2^k\). The paper is concluded with some remarks and directions for future work.
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6010508
zbMATH Keywords: cyclic codes
zbMATH Keywords: Gray map
zbMATH Keywords: codes over \(R_{k}\)
zbMATH Keywords: optimal codes
MaRDI profile type: Publication
full work available at URL: s10623-011-9539-4
OpenAlex ID: W2067264852
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DOI: 10.1007/S10623-011-9539-4
description: scientific article; zbMATH DE number 6010508