Classification of self-dual cyclic codes over the chain ring Zp[ u] / ⟨ u3

Boran Kim, Yoonjin Lee

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We classify all the cyclic self-dual codes of length pk over the finite chain ring R: = Zp[ u] / ⟨ u3⟩ , which is not a Galois ring, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over R of length pk for every prime p. We then prove that if a cyclic code over R of length pk is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over Z2[ u] / ⟨ u3⟩ of length 2 k. Finally, we obtain a mass formula for counting cyclic self-dual codes over Z2[ u] / ⟨ u3⟩ of length 2 k.

Original languageEnglish
Pages (from-to)2247-2273
Number of pages27
JournalDesigns, Codes, and Cryptography
Volume88
Issue number10
DOIs
StatePublished - 1 Oct 2020

Bibliographical note

Funding Information:
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)

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Chain ring
  • Cyclic code
  • Generator
  • Mass formula
  • Self-dual code
  • ideal

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