Entanglement-assisted quantum codes from Galois LCD codes

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the parameter of ebits $c$ is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on $k$-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits $c$ can be easily generated algebraically and not by computational search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.

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

[2]  Lina Zhang,et al.  On MDS linear complementary dual codes and entanglement-assisted quantum codes , 2018, Des. Codes Cryptogr..

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

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

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

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

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

[8]  Hanwu Chen,et al.  Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1 , 2016, Quantum Inf. Comput..

[9]  Yun Fan,et al.  Galois self-dual constacyclic codes , 2017, Des. Codes Cryptogr..

[10]  Xiusheng Liu,et al.  Quantum codes from linear codes over finite chain rings , 2017, Quantum Inf. Process..

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

[12]  Garry Bowen.  Entanglement required in achieving entanglement-assisted channel capacities , 2002 .

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

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

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

[16]  N. Sloane,et al.  Quantum error correction via codes over GF(4) , 1996, Proceedings of IEEE International Symposium on Information Theory.

[17]  Yun Fan,et al.  Galois LCD codes over finite fields , 2018, Finite Fields Their Appl..

[18]  James L. Massey,et al.  Linear codes with complementary duals , 1992, Discret. Math..

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

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

[21]  Mark M. Wilde,et al.  Duality in Entanglement-Assisted Quantum Error Correction , 2013, IEEE Transactions on Information Theory.