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

Boran Kim; Yoonjin Lee

doi:10.1007/s10623-020-00776-1

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

Boran Kim, Yoonjin Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We classify all the cyclic self-dual codes of length p^k over the finite chain ring R: = Z_p[ 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 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 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 language	English
Pages (from-to)	2247-2273
Number of pages	27
Journal	Designs, Codes, and Cryptography
Volume	88
Issue number	10
DOIs	https://doi.org/10.1007/s10623-020-00776-1
State	Published - 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

Access to Document

10.1007/s10623-020-00776-1

Cite this

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title = "Classification of self-dual cyclic codes over the chain ring Zp[ u] / ⟨ u3⟩",

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.",

keywords = "Chain ring, Cyclic code, Generator, Mass formula, Self-dual code, ideal",

author = "Boran Kim and Yoonjin Lee",

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: {\textcopyright} 2020, Springer Science+Business Media, LLC, part of Springer Nature.",

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N2 - 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.

AB - 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.

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