Optimal binary codes and binary construction of quantum codes

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.

Keywords

binary linear code / binary self-orthogonal code / quasi-cyclic code / Steane construction / quantum code

References

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

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

[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

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

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

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

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

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

[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

[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

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

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

[14] Aguilar-Melchor C, Gaborit P, KimJ 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

[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

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

[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

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

[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

[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

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

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

[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

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

[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

[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

[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

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

[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

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg