Quantum codes from cyclic codes over a class of nonchain rings

Lately, quantum error correcting codes have been considered as the Gray images of some special codes over various rings. In this study, we investigate the structure of the cyclic codes of arbitrary length over the finite nonchain ring R<sub>p</sub> = F<sub>p</sub>+vF<sub>p</sub>+···+v<sup>p-1</sup>F<sub>p</sub> where v<sup>p</sup> = v and obtain quantum error correcting codes over F<sub>p</sub> by using the Gray map defined in the work of Bayram and Siap (2014).

[1]  Santosh Kumar,et al.  Nonbinary Stabilizer Codes Over Finite Fields , 2005, IEEE Transactions on Information Theory.

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

[3]  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.

[4]  M. Özen,et al.  Quantum codes from codes over Gaussian integers with respect to the Mannheim metric , 2010 .

[5]  Pradeep Kiran Sarvepalli,et al.  Primitive Quantum BCH Codes over Finite Fields , 2006, 2006 IEEE International Symposium on Information Theory.

[6]  T. Aaron Gulliver,et al.  Quantum codes over rings , 2014 .

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

[8]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[9]  Fuzheng Gao,et al.  A modified weak Galerkin finite element method for a class of parabolic problems , 2014, J. Comput. Appl. Math..

[10]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[11]  I. Siap,et al.  Cyclic and constacyclic codes over a non-chain ring , 2014 .

[12]  Yasemin Cengellenmis,et al.  On Quantum Codes Obtained From Cyclic Codes Over F_2+vF_2+v^2F_2 , 2014, ArXiv.

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

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

[15]  Alexei E. Ashikhmin,et al.  Nonbinary quantum stabilizer codes , 2001, IEEE Trans. Inf. Theory.

[16]  Andrew M. Steane.  Enlargement of Calderbank-Shor-Steane quantum codes , 1999, IEEE Trans. Inf. Theory.

[17]  V. Tonchev.  The existence of optimal quaternary [28,20,6] and quantum [[28,12,6]] codes , 2014 .

[18]  Chung-Chin Lu,et al.  A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices , 2007, IEEE Transactions on Information Theory.