Complete Weight Distributions and MacWilliams Identities for Asymmetric Quantum Codes

In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators for qubit codes and then investigate the MacWilliams identities for these enumerators. Based on the generalized MacWilliams identities, we solve an open problem, namely, the Singleton-type bound for asymmetric quantum codes (AQCs). Besides, the Hamming-type and the first linear-programming-type bounds for the AQCs are deduced similarly.

[1]  Robert J. McEliece,et al.  New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities , 1977, IEEE Trans. Inf. Theory.

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

[3]  Eric M. Rains.  Quantum shadow enumerators , 1999, IEEE Trans. Inf. Theory.

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

[5]  Anuradha Sharma,et al.  The weight distribution of some irreducible cyclic codes , 2012, Finite Fields Their Appl..

[6]  Shudi Yang,et al.  Complete Weight Enumerators of a Class of Linear Codes From Weil Sums , 2019, IEEE Access.

[7]  Raymond Laflamme,et al.  Quantum Analog of the MacWilliams Identities for Classical Coding Theory , 1997 .

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

[9]  Pradeep Kiran Sarvepalli,et al.  Asymmetric quantum LDPC codes , 2008, 2008 IEEE International Symposium on Information Theory.

[10]  Viola,et al.  Theory of quantum error correction for general noise , 2000, Physical review letters.

[11]  J. Macwilliams.  A theorem on the distribution of weights in a systematic code , 1963 .

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

[13]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

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

[15]  Eric M. Rains.  Quantum Weight Enumerators , 1998, IEEE Trans. Inf. Theory.

[16]  Alexei Ashikhmin,et al.  Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators , 2016, IEEE Transactions on Information Theory.

[17]  Moe Z. Win,et al.  Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel , 2007, IEEE Transactions on Information Theory.

[18]  Shudi Yang,et al.  The weight enumerator of the duals of a class of cyclic codes with three zeros , 2015, Applicable Algebra in Engineering, Communication and Computing.

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

[20]  S. Litsyn,et al.  Upper bounds on the size of quantum codes , 1997, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[21]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[23]  Chaoping Xing,et al.  A new construction of quantum error-correcting codes , 2007 .

[24]  Yongjun Xu,et al.  Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups , 2016 .

[25]  P. Zanardi,et al.  Error avoiding quantum codes , 1997, quant-ph/9710041.

[26]  Simon Litsyn,et al.  Quantum error detection I: Statement of the problem , 1999, IEEE Trans. Inf. Theory.

[27]  P. Delsarte.  AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

[28]  Simon Litsyn,et al.  On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes , 1998, Des. Codes Cryptogr..

[29]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[30]  Richard Cleve.  Quantum stabilizer codes and classical linear codes , 1997 .

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

[32]  H. Makaruk,et al.  Generalized Noiseless Quantum Codes utilizing Quantum Enveloping Algebras , 1998, quant-ph/0003134.

[33]  Vladimir I. Levenshtein,et al.  Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces , 1995, IEEE Trans. Inf. Theory.

[34]  Nathan Linial,et al.  On the distance distribution of codes , 1995, IEEE Trans. Inf. Theory.

[35]  Min-Hsiu Hsieh,et al.  On the MacWilliams Identity for Classical and Quantum Convolutional Codes , 2014, IEEE Transactions on Communications.

[36]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[37]  Gerardo Vega,et al.  The Weight Distribution of an Extended Class of Reducible Cyclic Codes , 2012, IEEE Transactions on Information Theory.

[38]  Chengju Li,et al.  Weight distributions of cyclic codes with respect to pairwise coprime order elements , 2013, Finite Fields Their Appl..

[39]  E. Knill,et al.  Theory of quantum error-correcting codes , 1997 .

[40]  Chunming Tang,et al.  A construction of linear codes and their complete weight enumerators , 2017, Finite Fields Their Appl..

[41]  P. Zanardi,et al.  Noiseless Quantum Codes , 1997, quant-ph/9705044.

[42]  Chaoping Xing,et al.  Asymptotic bounds on quantum codes from algebraic geometry codes , 2006, IEEE Transactions on Information Theory.

[43]  Salah A. Aly,et al.  Asymmetric and Symmetric Subsystem BCH Codes and Beyond , 2008, ArXiv.

[44]  Felix Huber,et al.  Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity , 2017, ArXiv.

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

[46]  Shudi Yang,et al.  The weight distributions of two classes of p-ary cyclic codes with few weights , 2015, Finite Fields Their Appl..

[47]  Lei Hu,et al.  The weight distributions of two classes of p-ary cyclic codes , 2014, Finite Fields Their Appl..

[48]  Ekert,et al.  Quantum Error Correction for Communication. , 1996 .

[49]  Simon Litsyn,et al.  Quantum error detection II: Bounds , 1999, IEEE Trans. Inf. Theory.

[50]  Matti J. Aaltonen,et al.  A new upper bound on nonbinary block codes , 1990, Discret. Math..

[51]  Dongdai Lin,et al.  Complete weight enumerators of two classes of linear codes , 2017, Discret. Math..

[52]  Matti J. Aaltonen,et al.  Linear programming bounds for tree codes (Corresp.) , 1979, IEEE Trans. Inf. Theory.

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