Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 465–488 | Cite as

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

  • Sheng Wang
  • Yuan CaoEmail author
  • Yonglin Cao
Original Research


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.


Cyclic code Dual code Self-dual code Galois ring 

Mathematics Subject Classification

94B15 94B05 11T71 



This research is supported in part by the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007) and the National Natural Science Foundation of China (Grant Nos. 11671235, 11471255). Part of this work was done when Yonglin Cao was visiting the Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality.


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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShandong UniversityWeihaiChina
  2. 2.School of Mathematics and StatisticsShandong University of TechnologyZiboChina

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