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Classification of self-dual cyclic codes over the chain ring\(\mathbb Z_p[u]/\langle u^3 \rangle \)

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Abstract

We classify all the cyclic self-dual codes of length\(p^k\) over the finite chain ring\(\mathcal R:=\mathbb Z_p[u]/\langle u^3 \rangle \), which is not a Galois ring, wherep is a prime number andk is a positive integer. First, we find all the dual codes of cyclic codes over\({\mathcal R}\) of length\(p^k\) for every primep. We then prove that if a cyclic code over\({\mathcal R}\) of length\(p^k\) is self-dual, thenp should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over\(\mathbb Z_2[u]/\langle u^3 \rangle \) of length\(2^k\). Finally, we obtain a mass formula for counting cyclic self-dual codes over\(\mathbb Z_2[u]/\langle u^3 \rangle \) of length\(2^k\).

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Acknowledgements

We express our gratitude to the reviewers for their very helpful comments, which lead to improvement of the exposition of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Seobu-ro, Suwon, 16419, Republic of Korea

    Boran Kim

  2. Department of Mathematics, Ewha Womans University, Seoul, 03760, Republic of Korea

    Yoonjin Lee

Authors
  1. Boran Kim
  2. Yoonjin Lee

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Correspondence toYoonjin Lee.

Additional information

Communicated by J.-L. Kim.

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The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A1A01060467) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2016R1A5A1008055). Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574)

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Kim, B., Lee, Y. Classification of self-dual cyclic codes over the chain ring\(\mathbb Z_p[u]/\langle u^3 \rangle \).Des. Codes Cryptogr.88, 2247–2273 (2020). https://doi.org/10.1007/s10623-020-00776-1

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