Abstract
Let \({\mathbb {F}}_{3^m}\) be a finite field of cardinality \(3^m\), \(R={\mathbb {F}}_{3^m}[u]/\langle u^4\rangle \) which is a finite chain ring, and n be a positive integer satisfying \(\mathrm{gcd}(3,n)=1\). For any \(\delta ,\alpha \in {\mathbb {F}}_{3^m}^{\times }\), an explicit representation for all distinct \((\delta +\alpha u^2)\)-constacyclic codes over R of length 3n is given, formulas for the number of all such codes and the number of codewords in each code are provided, respectively. Moreover, the dual code for each of these codes is determined explicitly.
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References
Abualrub, T., Siap, I.: Cyclic codes over the ring \({\mathbb{Z}}_2+u{\mathbb{Z}}_2\) and \({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2\). Des. Codes Cryptogr. 42, 273–287 (2007)
Abualrub, T., Siap, I.: Constacyclic codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\). J. Franklin Inst. 346, 520–529 (2009)
Al-Ashker, M., Hamoudeh, M.: Cyclic codes over \(Z_2+uZ_2+u^2Z_2+\cdots +u^{k-1}Z_2\). Turk. J. Math. 35, 737–749 (2011)
Amerra, M.C.V., Nemenzo, F.R.: On \((1-u)\)-cyclic codes over \({\mathbb{F}}_{p^k}+u{\mathbb{F}}_{p^k}\). Appl. Math. Lett. 21, 1129–1133 (2008)
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.: On constacyclic codes over finite chain rings. Finite Fields Appl. 24, 124–135 (2013)
Cao, Y., Gao, Y.: Repeated root cyclic \({\mathbb{F}}_q\)-linear codes over \({\mathbb{F}}_{q^l}\). Finite Fields Appl. 31, 202–227 (2015)
Cao, Y., Cao, Y., Gao, J., Fu, F-W.: Constacyclic codes of length \(p^sn\) over \({\mathbb{F}_{p^m}+u{\mathbb{F}}}_{p^m}\). arXiv:1512.01406v1 (2015)
Dinh, H.Q.: Constacyclic codes of length \(2^s\) over Galois extension rings of \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inf. Theory 55, 1730–1740 (2009)
Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m}+u {\mathbb{F}}_{p^m}\). J. Algebra 324, 940–950 (2010)
Dinh, H.Q., Dhompongsa, 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., Li, C., Yue, Q.: Recent progress on weight distributions of cyclic codes over finite fields. J. Algebra Comb. Discrete Appl. 2, 39–63 (2015)
Dougherty, S.T., Gaborit, P., Harada, M., Solé, P.: Type II codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inf. Theory 45, 32–45 (1999)
Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)
Gulliver, T.A., Harada, M.: Construction of optimal type IV self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inf. Theory 45, 2520–2521 (1999)
Han, M., Ye, Y., Zhu, S., Xu, C., Dou, B.: Cyclic codes over \(R = F_p + uF_p +\cdots + u^{k-1}F_p\) with length \(p^sn\). Inf. Sci. 181, 926–934 (2011)
Huffman, W.C.: On the decomposition of self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\) with an automorphism of odd prime number. Finite Fields Appl. 13, 682–712 (2007)
Kai, X., Zhu, S., Li, P.: \((1+{\lambda {u}})\)-constacyclic codes over \({\mathbb{F}}_p[u]/\langle u^k\rangle \). J. Franklin Inst. 347, 751–762 (2010)
Norton, G., Sălăgean-Mandache, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000)
Qian, J.F., Zhang, L.N., Zhu, S.: \((1+u)\)-constacyclic and cyclic codes over \({{\mathbb{F}}}_2+u{{\mathbb{F}}}_2\) Appl. Math. Lett. 19, 820–823 (2006)
Singh, A.K., Kewat, P.K.: On cyclic codes over the ring \({\mathbb{Z}}_p[u]/\langle u^k\rangle \). Des. Codes Cryptogr. 72, 1–13 (2015)
Sobhani, R., Esmaeili, M.: Some constacyclic and cyclic codes over \({\mathbb{F}}_q[u]/\langle u^{t+1}\rangle \). IEICE Trans. Fundam. Electron. 93, 808–813 (2010)
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)
Wan, Z.-X.: Lectures on Finite Fields and Galois Rings. World Scientific Pub Co Inc., Singapore (2003)
Acknowledgements
We thank the anonymous referees for valuable comments that improved the presentation of this paper. Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11471255).
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Cao, Y., Cao, Y. & Dong, L. Complete classification of \((\delta +\alpha u^2)\)-constacyclic codes over \({\mathbb {F}}_{3^m}[u]/\langle u^4\rangle \) of length 3n . AAECC 29, 13–39 (2018). https://doi.org/10.1007/s00200-017-0328-9
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DOI: https://doi.org/10.1007/s00200-017-0328-9