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CYCLIC CODES OVER THE RING 𝔽p[u, v, w]/〈u2, v2, w2, uv - vu, vw - wv, uw - wu〉

  • Kewat, Pramod Kumar (Department of Applied Mathematics Indian Institute of Technology (ISM)) ;
  • Kushwaha, Sarika (Department of Applied Mathematics Indian Institute of Technology (ISM))
  • Received : 2016.10.26
  • Accepted : 2017.03.07
  • Published : 2018.01.31

Abstract

Let $R_{u{^2},v^2,w^2,p}$ be a finite non chain ring ${\mathbb{F}}_p[u,v,w]{\langle}u^2,\;v^2,\;w^2,\;uv-vu,\;vw-wv,\;uw-wu{\rangle}$, where p is a prime number. This ring is a part of family of Frobenius rings. In this paper, we explore the structures of cyclic codes over the ring $R_{u{^2},v^2,w^2,p}$ of arbitrary length. We obtain a unique set of generators for these codes and also characterize free cyclic codes. We show that Gray images of cyclic codes are 8-quasicyclic binary linear codes of length 8n over ${\mathbb{F}}_p$. We also determine the rank and the Hamming distance for these codes. At last, we have given some examples.

Table 1. Non zero cyclic codes of length 4 over Ru2,v2,w2,2.

E1BMAX_2018_v55n1_115_t0001.png 이미지

Table 2. Non zero cyclic codes of length 3 over Ru2,v2,w2,3.

E1BMAX_2018_v55n1_115_t0002.png 이미지

Table 3. Non zero cyclic codes of length 5 over Ru2,v2,w2,5.

E1BMAX_2018_v55n1_115_t0003.png 이미지

References

  1. T. Abualrub and I. Siap, 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 (2007), no. 3, 273-287. https://doi.org/10.1007/s10623-006-9034-5
  2. M. Al-Ashker and M. Hamoudeh, Cyclic codes over ${\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2+{\cdot}{\cdot}{\cdot}+u^{k-1}{\mathbb{Z}}_2$, Turkish J. Math. 35 (2011), no. 4, 737-749.
  3. A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2$, IEEE Trans. Inform. Theory 45 (1999), no. 4, 1250-1255. https://doi.org/10.1109/18.761278
  4. H. Q. Dinh, Constacylic codes of length $p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$, J. Algebra 324 (2010), no. 5, 940-950. https://doi.org/10.1016/j.jalgebra.2010.05.027
  5. H. Q. Dinh and S. 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
  6. S. T. Dougherty, S. Karadeniz, and B. Yildiz, Cyclic codes over $R_k$, Des. Codes Cryp-togr. 63 (2012), no. 1, 113-126. https://doi.org/10.1007/s10623-011-9539-4
  7. S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Inform. Theory 47 (2001), no. 1, 400-404. https://doi.org/10.1109/18.904544
  8. P. K. Kewat, B. Ghosh, and S. Pattanayak, Cyclic codes over the ring ${\mathbb{Z}}_p[u,\;v]/$, Finite Fields Appl. 34 (2015), 161-175. https://doi.org/10.1016/j.ffa.2015.01.005
  9. K. Shiromoto, Singleton bounds for codes over finite rings, J. Algebraic Combin. 12 (2000), no. 1, 95-99. https://doi.org/10.1023/A:1008767703006
  10. A. K. Singh and P. K. Kewat, Cyclic codes over ${\mathbb{Z}}_p[u]/$, Des. Codes Cryptogr. 74 (2015), no. 1, 1-13. https://doi.org/10.1007/s10623-013-9843-2
  11. R. Sobhani and M. Molakarimi, Some results on cyclic codes over the ring $R_{2,m}$, Turkish J. Math. 37 (2013), no. 6, 1061-1074. https://doi.org/10.3906/mat-1211-20
  12. B. Yildiz and S. Karadeniz, Cyclic codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2$, Des. Codes Cryptogr. 58 (2011), no. 3, 221-234. https://doi.org/10.1007/s10623-010-9399-3