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Cyclic codes over \({\mathbb {F}}_{2^m}[u]/\langle u^k\rangle \) of oddly even length

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Abstract

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 and \(R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}\) (\(u^k=0\)) where \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring \(R[x]/\langle x^{2n}-1\rangle \). In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Ashker, M., Hamoudeh, M.: Cyclic codes over \(Z_2+uZ_2+u^2Z_2+\ldots +u^{k-1}Z_2\). Tur. J. Math. 35(4), 737–749 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Bonnecaze, A., Udaya, P.: Cyclic codes and self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inform. Theory 45, 1250–1255 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dinh, H.Q.: Constacyclic codes of length \(2^s\) over Galois extension rings of \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inform. Theory 55, 1730–1740 (2009)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Norton, G., Sălăgean-Mandache, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra in Engrg. Comm. Comput. 10, 489–506 (2000)

    Article  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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 Key Basic Research Program of China (Grant No. 2013CB834204) and the National Natural Science Foundation of China (Grant No. 11471255).

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Cao, Y., Cao, Y. & Fu, FW. Cyclic codes over \({\mathbb {F}}_{2^m}[u]/\langle u^k\rangle \) of oddly even length. AAECC 27, 259–277 (2016). https://doi.org/10.1007/s00200-015-0281-4

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  • DOI: https://doi.org/10.1007/s00200-015-0281-4

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