Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON ℤpℤp[u]/k>-CYCLIC CODES AND THEIR WEIGHT ENUMERATORS

  • Received : 2019.07.30
  • Accepted : 2021.02.09
  • Published : 2021.05.01
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Abstract

In this paper we study the algebraic structure of ℤpℤp[u]/k>-cyclic codes, where uk = 0 and p is a prime. A ℤpℤp[u]/k>-linear code of length (r + s) is an Rk-submodule of ℤrp × Rsk with respect to a suitable scalar multiplication, where Rk = ℤp[u]/k>. Such a code can also be viewed as an Rk-submodule of ℤp[x]/r - 1> × Rk[x]/s - 1>. A new Gray map has been defined on ℤp[u]/k>. We have considered two cases for studying the algebraic structure of ℤpℤp[u]/k>-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤpℤp[u]/k>-linear codes. Examples have been given to construct ℤpℤp[u]/k>-cyclic codes, through which we get codes over ℤp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.

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Acknowledgement

The authors are thankful to the anonymous referee for his/her careful reading of the manuscript and helpful suggestions that greatly improved the final presentation of the manuscript. The second author would like to thank Ministry of Human Resource Development (MHRD), India, for providing financial support.

References

  1. I. Aydogdu, T. Abualrub, and I. Siap, On ℤ2ℤ2[u]-additive codes, Int. J. Comput. Math. 92 (2015), no. 9, 1806-1814. https://doi.org/10.1080/00207160.2013.859854
  2. I. Aydogdu and I. Siap, The structure of ℤ2ℤ2s-additive codes: bounds on the minimum distance, Appl. Math. Inf. Sci. 7 (2013), no. 6, 2271-2278. https://doi.org/10.12785/amis/070617
  3. I. Aydogdu and I. Siap, On ℤprℤps-additive codes, Linear Multilinear Algebra 63 (2015), no. 10, 2089-2102. https://doi.org/10.1080/03081087.2014.952728
  4. I. Aydogdu, I. Siap, and R. Ten-Valls, On the structure of ℤ2ℤ2[u3]-linear and cyclic codes, Finite Fields Appl. 48 (2017), 241-260. https://doi.org/10.1016/j.ffa.2017.03.001
  5. J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls, ℤ2ℤ4-additive cyclic codes, generator polynomials, and dual codes, IEEE Trans. Inform. Theory 62 (2016), no. 11, 6348-6354. https://doi.org/10.1109/TIT.2016.2611528
  6. J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls, ℤ2-double cyclic codes, Des. Codes Cryptogr. 86 (2018), no. 3, 463-479. https://doi.org/10.1007/s10623-017-0334-8
  7. L. Diao and J. Gao, ℤpℤp[u]-additive cyclic codes, Int. J. Inf. Coding Theory 5 (2018), no. 1, 1-17. https://doi.org/10.1504/IJICOT.2018.091815
  8. A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, The ℤ4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301-319. https://doi.org/10.1109/18.312154
  9. L. Qian and X. Cao, Bounds and optimal q-ary codes derived from the ℤqR-cyclic codes, IEEE Trans. Inform. Theory 66 (2020), no. 2, 923-935. https://doi.org/10.1109/TIT.2019.2927034
  10. J. Rifa and L. Ronquillo, Product perfect ℤ2ℤ4-linear codes in stegonography, International Symposium on Information Theory and Its Applications (2010), 696-701.
  11. M. Shi, R. Wu, and D. S. Krotov, On ℤpℤpk-additive codes and their duality, IEEE Trans. Inform. Theory 65 (2019), no. 6, 3841-3847. https://doi.org/10.1109/TIT.2018.2883759
  12. A. K. Singh and P. K. Kewat, On cyclic codes over the ring ℤp[u]/k>, Des. Codes Cryptogr. 74 (2015), no. 1, 1-13. https://doi.org/10.1007/s10623-013-9843-2