Pure Asymmetric Quantum MDS Codes from CSS Construction: A Complete Characterization

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

[1]  M. Mézard,et al.  Asymmetric quantum error-correcting codes , 2006, quant-ph/0606107.

[2]  Simeon Ball,et al.  On sets of vectors of a finite vector space in which every subset of basis size is a basis II , 2012, Designs, Codes and Cryptography.

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

[4]  Ron M. Roth,et al.  A construction of non-Reed-Solomon type MDS codes , 1989, IEEE Trans. Inf. Theory.

[5]  Neil J. A. Sloane,et al.  The theory of error-correcting codes (north-holland , 1977 .

[6]  Rudolf Lide,et al.  Finite fields , 1983 .

[7]  Markus Grassl,et al.  The Weights in MDS Codes , 2009, IEEE Transactions on Information Theory.

[8]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[9]  M. Rötteler,et al.  Asymmetric quantum codes: constructions, bounds and performance , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

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

[12]  Alexei E. Ashikhmin,et al.  Nonbinary quantum cyclic and subsystem codes over asymmetrically-decohered quantum channels , 2010, 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010, Cairo).

[13]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[14]  Simeon Ball.  ON LARGE SUBSETS OF A FINITE VECTOR SPACE IN WHICH EVERY SUBSET OF BASIS SIZE IS A BASIS , 2010 .

[15]  Chaoping Xing,et al.  Asymmetric Quantum Codes: Characterization and Constructions , 2010, IEEE Transactions on Information Theory.

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

[17]  J. H. Cole,et al.  Error correction optimisation in the presence of X/Z asymmetry , 2007, 0709.3875.