Some New Classes of Quantum MDS Codes From Constacyclic Codes

Quantum maximum-distance-separable (MDS) codes form an important family of quantum codes. In this paper, using Hermitian construction and classical constacyclic codes, we construct six classes of quantum MDS codes. Two of these six classes of quantum MDS codes have larger minimum distance than the ones available in the literature. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

[1]  Keqin Feng,et al.  Quantum codes [[6, 2, 3]]p and [[7, 3, 3]]p (p >= 3) exist , 2002, IEEE Trans. Inf. Theory.

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

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

[4]  Laflamme,et al.  Perfect Quantum Error Correcting Code. , 1996, Physical review letters.

[5]  Guanghui Zhang,et al.  Constructions of New Nonbinary Quantum Codes , 2015 .

[6]  Chaoping Xing,et al.  Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes , 2010, IEEE Transactions on Information Theory.

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

[8]  Guanghui Zhang,et al.  Application of Constacyclic Codes to Quantum MDS Codes , 2014, IEEE Transactions on Information Theory.

[9]  Y. Edel,et al.  Quantum twisted codes , 2000 .

[10]  Shixin Zhu,et al.  New quantum MDS codes derived from constacyclic codes , 2014, Quantum Inf. Process..

[11]  Zhuo Li,et al.  Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes , 2008, ArXiv.

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

[13]  Shixin Zhu,et al.  Constacyclic Codes and Some New Quantum MDS Codes , 2014, IEEE Transactions on Information Theory.

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

[15]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[16]  Dwijendra K. Ray-Chaudhuri,et al.  The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes , 2001, Des. Codes Cryptogr..

[17]  Shixin Zhu,et al.  New Quantum MDS Codes From Negacyclic Codes , 2013, IEEE Transactions on Information Theory.

[18]  Giuliano G. La Guardia,et al.  New Quantum MDS Codes , 2011, IEEE Transactions on Information Theory.

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

[20]  Chaoping Xing,et al.  A Construction of New Quantum MDS Codes , 2013, IEEE Transactions on Information Theory.

[21]  Martin Rötteler,et al.  On quantum MDS codes , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

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

[23]  Hao Chen,et al.  Quantum codes from concatenated algebraic-geometric codes , 2005, IEEE Transactions on Information Theory.

[24]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  Wenchao Cai,et al.  On self-dual constacyclic codes over finite fields , 2015, Des. Codes Cryptogr..

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