## Abstract

We classify all the cyclic self-dual codes of length p^{k} over the finite chain ring R: = Z_{p}[ u] / ⟨ u^{3}⟩ , 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 p^{k} for every prime p. We then prove that if a cyclic code over R of length p^{k} is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over Z_{2}[ u] / ⟨ u^{3}⟩ of length 2 ^{k}. Finally, we obtain a mass formula for counting cyclic self-dual codes over Z_{2}[ u] / ⟨ u^{3}⟩ of length 2 ^{k}.

Original language | English |
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Pages (from-to) | 2247-2273 |

Number of pages | 27 |

Journal | Designs, Codes, and Cryptography |

Volume | 88 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2020 |

## Keywords

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

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