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Optimal binary codes and binary construction of quantum codes

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Abstract

This paper discusses optimal binary codes and pure binary quantum codes created using Steane construction. First, a local search algorithm for a special subclass of quasi-cyclic codes is proposed, then five binary quasi-cyclic codes are built. Second, three classical construction methods are generalized for new codes from old such that they are suitable for constructing binary self-orthogonal codes, and 62 binary codes and six subcode chains of obtained self-orthogonal codes are designed. Third, six pure binary quantum codes are constructed from the code pairs obtained through Steane construction. There are 66 good binary codes that include 12 optimal linear codes, 45 known optimal linear codes, and nine known optimal self-orthogonal codes. The six pure binary quantum codes all achieve the performance of their additive counterparts constructed by quaternary construction and thus are known optimal codes.

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References

  1. Shannon C E. A mathematical theory of communication. Bell System Technical Journal, 1948, 27: 379–423, 623–656

    Article  MATH  MathSciNet  Google Scholar 

  2. Huffman W C, Pless V. Fundamentals of Error-Correcting Codes. Cambridge University Press, 2003

    Book  MATH  Google Scholar 

  3. Grassl M. Code tables: bounds on the parameters of various types of codes. http://www.codetables.de/

  4. Shor P W. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 1995, 52: 2493–2496

    Article  Google Scholar 

  5. Steane A M. Error correcting codes in quantum theory. Physical Review Letters, 1996, 77: 793–797

    Article  MATH  MathSciNet  Google Scholar 

  6. Calderbank A R, Shor P W. Good quantum error-correcting codes exist. Physical Review A, 1996, 54: 1098–1105

    Article  Google Scholar 

  7. Steane A M. Enlargement of calderbank-shor-steane quantum codes. IEEE Transactions on Information Theory, 1999, 45: 2492–2495

    Article  MATH  MathSciNet  Google Scholar 

  8. Bouyuklieva S. Some optimal self-orthogonal and self-dual codes. Discrete Mathematics, 2004, 287: 1–10

    Article  MATH  MathSciNet  Google Scholar 

  9. Bouyuklieva S, Anton M, Wolfgang W. Automorphisms of extremal codes. IEEE Transactions on Information Theory, 2010, 56(5): 2091–2096

    Article  Google Scholar 

  10. Bouyuklieva S, Ostergard P R J. New constructions of optimal self-dual binary codes of length 54. Designs, Codes and Cryptography, 2006, 41(1): 101–109

    Article  MATH  MathSciNet  Google Scholar 

  11. Bilous R.T. Enumeration of the binary self-dual codes of length 34. Journal of Combinatorial Mathematics and Combinatorial Computing, 2006, 59: 173–211

    MATH  MathSciNet  Google Scholar 

  12. Harada M, Munemasa A. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6(2): 229–235

    Article  MATH  MathSciNet  Google Scholar 

  13. Bouyuklieva S, Bouyukliev I. An algorithm for classification of binary self-dual codes. IEEE Transactions on Information Theory, 2012, 58(6): 3933–3940

    Article  MathSciNet  Google Scholar 

  14. Aguilar-Melchor C, Gaborit P, Kim J L, Sok L, Sole P, Vega G. Classification of extremal and s-extremal binary self-dual codes of length 38. IEEE Transactions on Information Theory, 2012, 58(4): 2253–2262

    Article  Google Scholar 

  15. Betsumiya K, Harada M, Munemasa A. A complete classification of doubly even self-dual codes of length 40. http://arxiv.org/abs/1104.3727v4, 2013.3

    Google Scholar 

  16. Ruihu Li, Xueliang Li. Binary construction of quantum codes of minimum distances five and six. Discrete Mathematics, 2008, 308: 1603–1611

    Article  MATH  MathSciNet  Google Scholar 

  17. Ruihu Li, Xueliang Li. Binary construction of quantum codes of minimum distance three and four. IEEE Transactions on Information Theory, 2004, 50(6): 1331–1335

    Article  MATH  Google Scholar 

  18. Sloane N A J. Is there a (72; 36) d = 16 self-dual code?. IEEE Transactions on Information Theory, 1973, 19(2): 251

    Article  MATH  MathSciNet  Google Scholar 

  19. Feulner T, Nebe G. The automorphism group of a self-dual binary [72; 36; 16] code does not contain Z7, Z3 × Z3, or D10. IEEE Transactions on Information Theory, 2012, 58(11): 6916–6924

    Article  MathSciNet  Google Scholar 

  20. O’Brien E A, Willems W. On the automorphism group of a binary selfdual doubly-even [72; 36; 16] code. IEEE Transactions on Information Theory, 2011, 57(7): 4445–4451

    Article  MathSciNet  Google Scholar 

  21. Daskalov R, Hristov P. New binary one-generator quasi-cyclic codes. IEEE Transactions on Information Theory, 2003, 49(11): 3001–3005

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen E Z. New quasi-cyclic codes from simplex codes. IEEE Transactions on Information Theory, 2007, 53(3): 1193–1196

    Article  Google Scholar 

  23. Grassl M, White G S. New codes from chains of quasi-cyclic codes. In: Proceedings of the 2005 IEEE International Symposium on Information Theory. 2005, 2095–2099

    Google Scholar 

  24. Lally K, Fitzpatrick P. Algebraic structure of quasicyclic codes. Discrete Applied Mathematics, 2001, 111: 157–175

    Article  MATH  MathSciNet  Google Scholar 

  25. Ling S, Sole P. On the algebraic structure of quasi-cyclic codes I: finite fields. 2005 IEEE International Sympium on Information Theory, 2001, 47(7): 2751–2759

    MATH  MathSciNet  Google Scholar 

  26. Cayrel P L, Chabot C, Necer A. Quasi-cyclic codes as codes over rings of matrices. Finite Fields and Their Applications, 2010, 16: 100–115

    Article  MATH  MathSciNet  Google Scholar 

  27. Cao Y L. 1-generator quasi-cyclic codes over finite chain rings. Applicable Algebra in Engineering, Communication and Computing, 2013, 24(1): 53–72

    Article  MATH  MathSciNet  Google Scholar 

  28. Cao Y L, Gao J. Constructing quasi-cyclic codes from linear algebra theory. Designs, Codes and Cryptography, 2013, 67(1): 59–75

    Article  MATH  MathSciNet  Google Scholar 

  29. Calderbank A R, Rains E M, Shor P W, Sloane N J A. Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory, 1998, 44: 1369–1387

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Electronics and Information, Northwestern Polytechnical University, Xi’an, 710072, China

    Weiliang Wang & Yangyu Fan

  2. Science College, Air Force Engineering University, Xi’an, 710051, China

    Weiliang Wang & Ruihu Li

Authors
  1. Weiliang Wang
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  2. Yangyu Fan
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  3. Ruihu Li
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Corresponding author

Correspondence to Weiliang Wang.

Additional information

Weiliang Wang received his BS and MS degrees from the National University of Defense Technology, China in 1998 and Peking University, China in 2006, respecfively. Wang is a PhD candidate in Northwestern Polytechnical University, China. His research interests include quantum coding and data mining.

Yangyu Fan is a professor in the School of Electronics and Information, Northwestern Polytechnical University. He received his PhD degree from Northwestern Polytechnical University, China in 1999. His research interests include quantum communication, optical wireless communication, and image processing.

Ruihu Li is a professor in the Science College, Air Force Engineering University, China. He received his PhD degree from Northwestern Polytechnical University, China in 2004. His research interests include algebraic coding, quantum coding, and cryptography.

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Wang, W., Fan, Y. & Li, R. Optimal binary codes and binary construction of quantum codes. Front. Comput. Sci. 8, 1024–1031 (2014). https://doi.org/10.1007/s11704-014-3469-z

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  • DOI: https://doi.org/10.1007/s11704-014-3469-z

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