Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SR-ADDITIVE CODES

  • Received : 2018.10.18
  • Accepted : 2018.12.18
  • Published : 2019.09.30
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Abstract

In this paper, we introduce SR-additive codes as a generalization of the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes, where S is an R-algebra and an SR-additive code is an R-submodule of $S^{\alpha}{\times}R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes are generalized to SR-additive codes. Also the singleton bound for SR-additive codes and some results on one weight SR-additive codes are given. Among other important results, we obtain the structure of SR-additive cyclic codes. As some results of the theory, the structure of cyclic ${\mathbb{Z}}_2{\mathbb{Z}}_4$, ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$, ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$ and $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$-additive codes are presented.

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References

  1. T. Abualrub and I. Siap, Cyclic codes over the rings $Z_2+uZ_2$ and $Z_2+uZ_2+u^2Z_2$, Des. Codes Cryptogr. 42 (2007), no. 3, 273-287. https://doi.org/10.1007/s10623-006-9034-5
  2. T. Abualrub, I. Siap, and N. Aydin, ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes, IEEE Trans. Inform. Theory 60 (2014), no. 3, 1508-1514. https://doi.org/10.1109/TIT.2014.2299791
  3. I. Aydogdu, T. Abualrub, and I. Siap, On ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes, Int. J. Comput. Math. 92 (2015), no. 9, 1806-1814. https://doi.org/10.1080/00207160.2013.859854
  4. I. Aydogdu and I. Siap, The structure of ${\mathbb{Z}}_2{\mathbb{Z}}_2s$-additive codes: bounds on the minimum distance, Appl. Math. Inf. Sci. 7 (2013), no. 6, 2271-2278. https://doi.org/10.12785/amis/070617
  5. I. Aydogdu and I. Siap, On ${\mathbb{Z}}_pr{\mathbb{Z}}_ps$-additive codes, Linear Multilinear Algebra 63 (2015), no. 10, 2089-2102. https://doi.org/10.1080/03081087.2014.952728
  6. J. J. Bernal, J. Borges, C. Fernandez-Cordoba, and M. Villanueva, Permutation decoding of ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear codes, Des. Codes Cryptogr. 76 (2015), no. 2, 269-277. https://doi.org/10.1007/s10623-014-9946-4
  7. M. Bilal, J. Borges, S. T. Dougherty, and C. Fernandez-Cordoba, Maximum distance separable codes over ${\mathbb{Z}}_4$ and ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_4$, Des. Codes Cryptogr. 61 (2011), no. 1, 31-40. https://doi.org/10.1007/s10623-010-9437-1
  8. J. Borges and C. Fernandez-Cordoba, There is exactly one ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic 1-perfect code, Des. Codes Cryptogr. 85 (2017), no. 3, 557-566. https://doi.org/10.1007/s10623-016-0323-3
  9. J. Borges, C. Fernandez-Cordoba, J. Pujol, J. Rifa, and M. Villanueva, ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear codes: generator matrices and duality, Des. Codes Cryptogr. 54 (2010), no. 2, 167-179. https://doi.org/10.1007/s10623-009-9316-9
  10. J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls, On ${\mathbb{Z}}_pr{\mathbb{Z}}_ps$-additive cyclic codes, (2016).
  11. J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls, Linear and cyclic codes over direct product of chain rings, Math. Meth. Appl. Sci. (2017).
  12. H. Q. Dinh and S. R. Lopez-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory 50 (2004), no. 8, 1728-1744. https://doi.org/10.1109/TIT.2004.831789
  13. S. T. Dougherty and C. Fernandez-Cordoba, ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive formally self-dual codes, Des. Codes Cryptogr. 72 (2014), no. 2, 435-453. https://doi.org/10.1007/s10623-012-9773-4
  14. S. T. Dougherty, H. Liu, and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011), 39-53.
  15. S. T. Dougherty, H. Liu, and L. Yu, One weight ${\mathbb{Z}}_2{\mathbb{Z}}_4$ additive codes, Appl. Algebra Engrg. Comm. Comput. 27 (2016), no. 2, 123-138. https://doi.org/10.1007/s00200-015-0273-4
  16. C. Fernandez-Cordoba, J. Pujol, and M. Villanueva, ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear codes: rank and kernel, Des. Codes Cryptogr. 56 (2010), no. 1, 43-59. https://doi.org/10.1007/s10623-009-9340-9
  17. J. Rifa, F. I. Solov'eva, and M. Villanueva, On the intersection of ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive perfect codes, IEEE Trans. Inform. Theory 54 (2008), no. 3, 1346-1356. https://doi.org/10.1109/TIT.2007.915917
  18. K. Samei and M. R. Alimoradi, Cyclic codes over the ring $F_2+uF_2+vF_2$, Comput. Appl. Math. 37 (2018), no. 3, 2489-2502. https://doi.org/10.1007/s40314-017-0460-y
  19. K. Samei and S. Mahmoudi, Cyclic R-additive codes, Discrete Math. 340 (2017), no. 7, 1657-1668. https://doi.org/10.1016/j.disc.2016.11.007
  20. K. Samei and S. Mahmoudi, Singleton bounds for R-additive codes, Adv. Math. Commun. 12 (2018), no. 1, 107-114. https://doi.org/10.3934/amc.2018006
  21. K. Samei and S. Sadeghi, Maximum distance separable codes over ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2s$, J. Algebra Appl. 17 (2018), no. 7, 1850136, 12 pp. https://doi.org/10.1142/S0219498818501360
  22. B. Srinivasulu and M. Bhaintwal, ${\mathbb{Z}}_2({\mathbb{Z}}_2+u{\mathbb{Z}}_2)$-additive cyclic codes and their duals, Discrete Math. Algorithms Appl. 8 (2016), no. 2, 1650027, 19 pp. https://doi.org/10.1142/S1793830916500270
  23. J. A. Wood, The structure of linear codes of constant weight, Trans. Amer. Math. Soc. 354 (2002), no. 3, 1007-1026. https://doi.org/10.1090/S0002-9947-01-02905-1