Quantum codes from linear codes over finite chain rings

In this paper, we provide two methods of constructing quantum codes from linear codes over finite chain rings. The first one is derived from the Calderbank–Shor–Steane (CSS) construction applied to self-dual codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the linear codes over finite chain ring $${\mathbb {F}}_{p^{2m}}+u{\mathbb {F}}_{p^{2m}}$$Fp2m+uFp2m. The good parameters of quantum codes from cyclic codes over finite chain rings are obtained.

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

[2]  Pradeep Kiran Sarvepalli,et al.  On Quantum and Classical BCH Codes , 2006, IEEE Transactions on Information Theory.

[3]  Hongwei Liu,et al.  Self-dual codes over commutative Frobenius rings , 2010, Finite Fields Their Appl..

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

[5]  Guanghui Zhang,et al.  Enumeration formulas for self-dual cyclic codes , 2016, Finite Fields Their Appl..

[6]  A. Sălăgean,et al.  On the stucture of linear and cyclic codes over finite chain rings , 2018 .

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

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

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

[10]  Graham H. Norton,et al.  On the Structure of Linear and Cyclic Codes over a Finite Chain Ring , 2000, Applicable Algebra in Engineering, Communication and Computing.

[11]  Shixin Zhu,et al.  New quantum codes from dual-containing cyclic codes over finite rings , 2016, Quantum Inf. Process..

[12]  Hongwei Liu,et al.  Independence of vectors in codes over rings , 2009, Des. Codes Cryptogr..

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

[14]  Sergio R. López-Permouth,et al.  Cyclic and negacyclic codes over finite chain rings , 2004, IEEE Transactions on Information Theory.

[15]  Xiusheng Liu,et al.  LCD codes over finite chain rings , 2015, Finite Fields Their Appl..

[16]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[17]  N. Sloane,et al.  Quantum Error Correction Via Codes Over GF , 1998 .