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, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over \({\mathcal R}\) of length \(p^k\) for every prime p. We then prove that if a cyclic code over \({\mathcal R}\) 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 \(\mathbb Z_2[u]/\langle u^3 \rangle \) 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 \(2^k\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abualrub T., Oehmke R.B.: On the generator of \(\mathbb{Z}_{4}\) cyclic codes of length \(2^{e}\). IEEE Trans. Inf. Theory 49, 2126–2133 (2003).
Abualrub T., Siap I.: Cyclic codes over the rings \(\mathbb{Z}_2+u\mathbb{Z}_2\) and \(\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2\). Des. Codes Cryptogr. 42, 273–287 (2007).
Al-Ashker M., Hamoudeh M.: Cyclic codes over \(\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2+\cdots +u^{k-1}\mathbb{Z}_2\). Turk. J. Math. 35, 737–7494 (2011).
Bonnecaze A., Udaya P.: Cyclic codes and self-dual codes over \(\mathbb{F}_2+u\mathbb{F}_2\). IEEE Trans. Inf. Theory 45, 1250–1255 (1999).
Cao Y., Fu F.-W.: Cyclic codes over \(\mathbb{F}_{2^m}[u]/\langle u^k \rangle \) of oddly even length. Appl. Algebra Eng. Commun. Comput. 27, 259–277 (2016).
Cao Y., Li Q.: Cyclic codes of odd length over \({\mathbb{Z}}_4[u]/\langle u^k \rangle \). Cryptogr. Commun. 9, 599–624 (2017).
Dinh H.Q.: Constacyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}\). J. Algebra 5, 940–950 (2010).
Dinh H.Q., Dohmpongsa S., Sriboonchitta S.: Repeated-root constacyclic codes of prime power length over \(\frac{{\mathbb{F}}_{p^m}[u]}{\langle u^a \rangle }\) and their duals. Discrete Math. 339, 1706–1715 (2016).
Dinh H.Q., Fan Y., Liu H., Liu X., Sriboonchitta S.: On self-dual constacyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). Discrete Math. 341, 324–335 (2018).
Dinh H.Q., López-Permouth S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50, 1728–1744 (2004).
Dinh H.Q., Singh A.K., Kumar P., Sriboonchitta S.: On the structure of cyclic codes over the ring \({\mathbb{Z}}_{2^s}[u]/\langle u^k \rangle \). Discrete Math. 341, 2243–2275 (2018).
Dinh H.Q., Wang L., Zhu S.: Negacyclic codes of length \(2p^s\) over \({\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}\). Finite Fields Appl. 31, 178–201 (2015).
Dougherty S.T., Karadeniz S., Yildiz B.: Cyclic codes over \(R_k\). Des. Codes Cryptogr. 63, 113–126 (2012).
Greferath M., Schmidt S.E.: Gray isometries for finite chain rings and a nonlinear ternary \((36, 3^{12}, 15)\) code. IEEE Trans. Inf. Theory 45, 2522–2524 (1999).
Hammons Jr. A.R., Kummar P.V., Calderbank A.R., Sloane N.J.A., Sole P.: The \({\mathbb{Z}}_4\) -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).
Kiah H.M., Leung K.H., Ling S.: Cyclic codes over \(GR(p^2, m)\) of length \(p^k\). Finite Fields Appl. 14, 834–846 (2008).
Kiah H.M., Leung K.H., Ling S.: A note on cyclic codes over \(GR(p^2, m)\) of length \(p^k\). Des. Codes Cryptogr. 63, 105–112 (2012).
Kim B., Lee Y.: Construction of extremal self-dual codes over \({\mathbb{Z}}_8\) and \({\mathbb{Z}}_{16}\). Des. Codes Cryptogr. 81, 239–257 (2016).
Kim B., Lee Y.: Lee weights of cyclic self-dual codes over Galois rings of characteristic \(p^2\). Finite Fields Appl. 45, 107–130 (2017).
Kim B., Lee Y.: A mass formula for cyclic codes over Galois rings of characteristic \(p^3\). Finite Fields Appl. 52, 214–242 (2018).
Kim B., Lee Y., Doo J.: Classification of cyclic codes over a non-Galois chain ring \({\mathbb{Z}}_p[u]/\langle u^3 \rangle \). Finite Fields Appl. 59, 208–237 (2019).
Singh A.K., Kewat P.K.: On cyclic codes over the ring \(\mathbb{Z}_p[u]/\langle u^k \rangle \). Des. Codes Cryptogr. 74, 1–13 (2015).
Sobhani R.: Complete classification of \((\delta +\alpha u^2)\) -constacyclic codes of length \(p^k\) over \(\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}\). Finite Fields Appl. 34, 123–138 (2015).
Sobhani R., Esmaeili M.: A note on cyclic codes over \(GR(p^2, m)\) of length \(p^k\). Finite Fields Appl. 15, 387–391 (2009).
Wolfmann J.: Binary images of cyclic codes over \({\mathbb{Z}}_4\). IEEE Trans. Inf. Theory 47, 1773–1779 (2001).
Acknowledgements
We express our gratitude to the reviewers for their very helpful comments, which lead to improvement of the exposition of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.-L. Kim.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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)
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-020-00776-1