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Published in: Journal of Applied Mathematics and Computing 1-2/2019

11-05-2018 | Original Research

Constructing self-dual cyclic codes over \({\mathbb {Z}}_{9}\) of length 3n

Authors: Sheng Wang, Yuan Cao, Yonglin Cao

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2019

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Abstract

In this paper, we study cyclic codes over \(\mathbb {Z}_9\) of length 3n, where n is a positive integer satisfying \(\mathrm{gcd}(3,n)=1\). First, a canonical form decomposition of any cyclic code over \(\mathbb {Z}_9\) of length 3n are given and a unique set of generators for each subcode is presented. Hence the structure of any cyclic code over \(\mathbb {Z}_9\) of length 3n is determined. From this decomposition, formulas for the number of all codes and the number of codewords in each code are given. Then dual codes and self-duality of these codes are investigated. As an application, all 10061824 distinct cyclic codes over \(\mathbb {Z}_9\) of length 24 and all 544 self-dual codes among them are listed explicitly. Moreover, 280 new and good self-dual cyclic codes over \(\mathbb {Z}_9\) with basic parameters \(\left( 24, 3^{24}, 3\right) \) are obtained.

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Appendix
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Metadata
Title
Constructing self-dual cyclic codes over of length 3n
Authors
Sheng Wang
Yuan Cao
Yonglin Cao
Publication date
11-05-2018
Publisher
Springer Berlin Heidelberg
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2019
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-018-1188-6

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