On the codes over the Z_3+vZ_3+v^2Z_3

In this paper, we study the structure of cyclic, quasi cyclic, constacyclic codes and their skew codes over the finite ring R. The Gray images of cyclic, quasi cyclic, skew cyclic, skew quasi cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. Some examples are given. Firstly, quasi constacyclic and skew quasi constacyclic codes are introduced. By giving two inner product, it is investigated their duality. A sufficient condition for 1 generator skew quasi constacyclic codes to be free is determined.

[1]  Shixin Zhu,et al.  (1+u) constacyclic and cyclic codes over F2+uF2 , 2006, Appl. Math. Lett..

[2]  Irfan Siap,et al.  Structure of codes over the ring Z3[v]/v3vZ3[v]/〈v3−v〉 , 2013, Applicable Algebra in Engineering, Communication and Computing.

[3]  T. Beth,et al.  On optimal quantum codes , 2003, quant-ph/0312164.

[4]  Patanee Udomkavanich,et al.  Skew constacyclic codes over finite chain rings , 2010, Adv. Math. Commun..

[5]  Maheshanand Bhaintwal,et al.  On Quasi-cyclic Codes over Integer Residue Rings , 2007, AAECC.

[6]  Irfan Siap,et al.  Structure of codes over the ring Z3[v]/(v3-v) , 2013, Appl. Algebra Eng. Commun. Comput..

[7]  Steven T. Dougherty,et al.  Codes over an infinite family of rings with a Gray map , 2014, Des. Codes Cryptogr..

[8]  Patrick Solé,et al.  On the Algebraic Structure of Quasi-cyclic Codes II: Chain Rings , 2003, Des. Codes Cryptogr..

[9]  Patrick Solé,et al.  On the algebraic structure of quasi-cyclic codes III: generator theory , 2005, IEEE Transactions on Information Theory.

[10]  Jianfa Qian Quantum Codes from Cyclic Codes over $F_2+vF_2$ , 2013 .

[11]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[13]  N. Aydin,et al.  On θ-cyclic codes over 𝔽 2 + v𝔽 2 . , 2012 .

[14]  Patrick Solé,et al.  On the algebraic structure of quasi-cyclic codes I: Finite fields , 2001, IEEE Trans. Inf. Theory.

[15]  Ali Ghrayeb,et al.  On the Construction of Skew Quasi-Cyclic Codes , 2008, IEEE Transactions on Information Theory.

[16]  Taher Abualrub,et al.  Skew cyclic codes of arbitrary length , 2011, Int. J. Inf. Coding Theory.

[17]  Patrick Solé,et al.  Skew constacyclic codes over Galois rings , 2008, Adv. Math. Commun..

[18]  Taher Abualrub,et al.  On θ-cyclic codes over F2 + vF2 , 2012 .

[19]  Mohammad Ashraf,et al.  Quantum codes from cyclic codes over F3 + vF3 , 2014 .

[20]  Steane,et al.  Simple quantum error-correcting codes. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[21]  Jian Gao,et al.  Skew Generalized Quasi-Cyclic Codes over Finite Fields , 2013, ArXiv.

[22]  Wenping Ma,et al.  QUANTUM CODES FROM CYCLIC CODES OVER FINITE RING , 2009 .

[23]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[24]  Felix Ulmer,et al.  Skew-cyclic codes , 2006, Applicable Algebra in Engineering, Communication and Computing.

[25]  Minjia Shi,et al.  Cyclic codes over F2 + vF2 , 2009, 2009 IEEE International Symposium on Information Theory.

[26]  Felix Ulmer,et al.  Coding with skew polynomial rings , 2009, J. Symb. Comput..

[27]  Jian Gao SKEW CYCLIC CODES OVER Fp+ vFp , 2013 .

[28]  Mingzhong Wu Skew Cyclic and Quasi-Cyclic Codes of Arbitrary Length over Galois Rings , 2013 .

[29]  Yasemin Cengellenmis,et al.  On quantum codes obtained from cyclic codes over A2 , 2015 .

[30]  Maheshanand Bhaintwal Skew quasi-cyclic codes over Galois rings , 2012, Des. Codes Cryptogr..

[31]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[32]  Quantum codes over the ring F_2 + uF_2 + u^2F_2 + ... + u^mF_2 , 2015 .

[33]  Maheshanand Bhaintwal,et al.  On quasi-cyclic codes over Zq , 2009, Applicable Algebra in Engineering, Communication and Computing.

[34]  Wenping Ma,et al.  Gray Map and Quantum Codes over the Ring F_2+uF_2+u^2F_2 , 2011, 2011IEEE 10th International Conference on Trust, Security and Privacy in Computing and Communications.