Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1

The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

[1]  Martin Rötteler,et al.  Quantum MDS codes over small fields , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

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

[3]  Vladimir D. Tonchev,et al.  A Characterization of Entanglement-Assisted Quantum Low-Density Parity-Check Codes , 2011, IEEE Transactions on Information Theory.

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

[5]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[6]  Dilip V. Sarwate,et al.  Pseudocyclic maximum- distance-separable codes , 1990, IEEE Trans. Inf. Theory.

[7]  T. Brun,et al.  Optimal entanglement formulas for entanglement-assisted quantum coding , 2008, 0804.1404.

[8]  Igor Devetak,et al.  Correcting Quantum Errors with Entanglement , 2006, Science.

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

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

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

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

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

[14]  Markus Grassl,et al.  Quantum Reed-Solomon Codes , 1999, AAECC.

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

[16]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[17]  Li-Yi Hsu,et al.  High Performance Entanglement-Assisted Quantum LDPC Codes Need Little Entanglement , 2009, IEEE Transactions on Information Theory.

[18]  Zunaira Babar,et al.  Entanglement-Assisted Quantum Turbo Codes , 2010, IEEE Transactions on Information Theory.

[19]  Vladimir D. Tonchev,et al.  Entanglement-assisted quantum low-density parity-check codes , 2010, ArXiv.

[20]  Zongben Xu,et al.  On [[n,n-4,3]]q Quantum MDS Codes for odd prime power q , 2009, ArXiv.

[21]  R. Schumann.  Quantum Information Theory , 2000, quant-ph/0010060.

[22]  Ruihu Li,et al.  Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes , 2014 .

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

[24]  I. Devetak,et al.  Entanglement-assisted quantum quasicyclic low-density parity-check codes , 2008, 0803.0100.

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

[26]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

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

[28]  Robert B. Griffiths,et al.  Quantum Error Correction , 2011 .

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