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SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER 𝔽pm+u𝔽pm+u2𝔽pm

  • Liu, Xiusheng (School of Mathematics and Physics Hubei Polytechnic University) ;
  • Xu, Xiaofang (School of Mathematics and Physics Hubei Polytechnic University)
  • Received : 2013.12.25
  • Published : 2014.07.01

Abstract

Constacyclic codes of length $p^s$ over $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{<x^{p^s}-1>}$. In this paper, we investigate constacyclic codes of length $p^s$ over R. The units of the ring R are of the forms ${\gamma}$, ${\alpha}+u{\beta}$, ${\alpha}+u{\beta}+u^2{\gamma}$ and ${\alpha}+u^2{\gamma}$, where ${\alpha}$, ${\beta}$ and ${\gamma}$ are nonzero elements of $\mathbb{F}_{p^m}$. We obtain the structures and Hamming distances of all (${\alpha}+u{\beta}$)-constacyclic codes and (${\alpha}+u{\beta}+u^2{\gamma}$)-constacyclic codes of length $p^s$ over R. Furthermore, we classify all cyclic codes of length $p^s$ over R, and by using the ring isomorphism we characterize ${\gamma}$-constacyclic codes of length $p^s$ over R.

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References

  1. T. Abulrub and R. Oehmke, On the generators of ${\mathbb{Z}}_4$ cyclic codes of length $2^e$, IEEE Trans. Inform. Theory 49 (2003), 2126-2133. https://doi.org/10.1109/TIT.2003.815763
  2. T. Blackford, Cyclic code over ${\mathbb{Z}}_4$ of oddly even length, Discrete Appl. Math. 138 (2003), no. 1, 27-40.
  3. T. Blackford, Negacyclic codes over ${\mathbb{Z}}_4$ of even length, IEEE Trans. Inform. Theory 49 (2003), no. 6, 1417-1424. https://doi.org/10.1109/TIT.2003.811915
  4. H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4252-4262. https://doi.org/10.1109/TIT.2005.859284
  5. H. Q. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Field Appl. 14 (2008), no. 1, 22-40. https://doi.org/10.1016/j.ffa.2007.07.001
  6. H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois exlension rings of ${\mathbb{F}}_2\;+\;u{\mathbb{F}}_2$, IEEE Trans. Inform. Theory 55 (2009), no. 4, 1730-1740. https://doi.org/10.1109/TIT.2009.2013015
  7. H. Q. Dinh, Constacyclic 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
  8. 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
  9. S. T. Dougherty and S. Ling, Cyclic codes over ${\mathbb{Z}}_4$ of even length, Des. Codes Cryptogr. 39 (2006), no. 2, 127-153. https://doi.org/10.1007/s10623-005-2773-x
  10. A. R. Hammous, Jr., P. V. Kumar, A. R. Calderbark, J. A. Sloame, and P. Sole, The ${\mathbb{Z}}_4$-linearity of Kordock, Preparata, Goethals, and releted codes, IEEE Trans. Inform. Theory 40 (1994), 301-319. https://doi.org/10.1109/18.312154
  11. W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
  12. P. Kanwar and S. R. Lopez-Permouth, Cyclic codes over the integers modulo ${\mathbb{Z}}_{p^m}$, Finite Field Appl. 3 (1997), no. 4, 334-352. https://doi.org/10.1006/ffta.1997.0189
  13. S. Ling, H. Niederreiter, and P. Sole, On the algebraic structure of quasi-cyclic codes. IV, Repeated root, Des. Codes. Cryplogr. 38 (2006), no. 3, 337-361. https://doi.org/10.1007/s10623-005-1431-7
  14. G. H. Norton and A. Salagean, On the struture of linear and cyclic codes over a finite chain ring, AAECC 10 (2000), no. 6, 489-506. https://doi.org/10.1007/PL00012382
  15. V. Pless and W. C. Huffman, Handbook of Coding Theory, Elsevier, Amsterdam, 1998.
  16. A. Salagean, Repelated-root cyclic and negacyclic codes over finite chain rings, Discrete Appl. Math. 154 (2006), 413-419. https://doi.org/10.1016/j.dam.2005.03.016
  17. J. Wolfmann, Negacyclic and cyclic codes over ${\mathbb{Z}}_4$, IEEE Trans. Inform. Theory. 45 (1999), no. 7, 2527-2532. https://doi.org/10.1109/18.796397
  18. S. Zhu and X. Kai, Dual and self-dual negacyclic codes of even length over ${\mathbb{Z}}_2a$, Discrete Math. 309 (2009), no. 8, 2382-2391. https://doi.org/10.1016/j.disc.2008.05.013
  19. S. Zhu and X. Kai, A class of constacyclic codes over ${\mathbb{Z}}_{p^m}$, Finite Field Appl. 16 (2010), no. 4, 243-254. https://doi.org/10.1016/j.ffa.2010.03.003