Optimal binary codes and binary construction of quantum codes

doi:10.1007/s11704-014-3469-z

Front. Comput. Sci.

2014, Vol. 8

Issue (6) : 1024-1031 https://doi.org/10.1007/s11704-014-3469-z

RESEARCH ARTICLE

Optimal binary codes and binary construction of quantum codes

Weiliang WANG^1,^2,^*(

),Yangyu FAN¹,Ruihu LI²

1. School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China
2. Science College, Air Force Engineering University, Xi’an 710051, China

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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.

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

Corresponding Author(s): Weiliang WANG

Issue Date: 27 November 2014

Cite this article:

Weiliang WANG,Yangyu FAN,Ruihu LI. Optimal binary codes and binary construction of quantum codes[J]. Front. Comput. Sci., 2014, 8(6): 1024-1031.

URL:

https://academic.hep.com.cn/fcs/EN/10.1007/s11704-014-3469-z
https://academic.hep.com.cn/fcs/EN/Y2014/V8/I6/1024

1	Shannon C E. A mathematical theory of communication. Bell System Technical Journal, 1948, 27: 379-423, 623-656 https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
2	Huffman W C, Pless V. Fundamentals of Error-Correcting Codes. Cambridge University Press, 2003 https://doi.org/10.1017/CBO9780511807077
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 https://doi.org/10.1103/PhysRevA.52.R2493
5	Steane A M. Error correcting codes in quantum theory. Physical Review Letters, 1996, 77: 793-797 https://doi.org/10.1103/PhysRevLett.77.793
6	Calderbank A R, Shor P W. Good quantum error-correcting codes exist. Physical Review A, 1996, 54: 1098-1105 https://doi.org/10.1103/PhysRevA.54.1098
7	Steane A M. Enlargement of calderbank-shor-steane quantum codes. IEEE Transactions on Information Theory, 1999, 45: 2492-2495 https://doi.org/10.1109/18.796388
8	Bouyuklieva S. Some optimal self-orthogonal and self-dual codes. Discrete Mathematics, 2004, 287: 1-10 https://doi.org/10.1016/j.disc.2004.06.010
9	Bouyuklieva S, Anton M, Wolfgang W. Automorphisms of extremal codes. IEEE Transactions on Information Theory, 2010, 56(5): 2091-2096 https://doi.org/10.1109/TIT.2010.2043763
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 https://doi.org/10.1007/s10623-006-0018-2
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 https://doi.org/10.3934/amc.2012.6.229
13	Bouyuklieva S, Bouyukliev I. An algorithm for classification of binary self-dual codes. IEEE Transactions on Information Theory, 2012, 58(6): 3933-3940 https://doi.org/10.1109/TIT.2012.2190134
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 https://doi.org/10.1109/TIT.2011.2177809
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 https://doi.org/10.1016/j.disc.2007.04.016
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 https://doi.org/10.1109/TIT.2004.828149
18	Sloane N A J. Is there a (72; 36) d= 16 self-dual code? IEEE Transactions on Information Theory, 1973, 19(2): 251 https://doi.org/10.1109/TIT.1973.1054975
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 https://doi.org/10.1109/TIT.2012.2208176
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 https://doi.org/10.1109/TIT.2011.2145850
21	Daskalov R, Hristov P. New binary one-generator quasi-cyclic codes. IEEE Transactions on Information Theory, 2003, 49(11): 3001-3005 https://doi.org/10.1109/TIT.2003.819337
22	Chen E Z. New quasi-cyclic codes from simplex codes. IEEE Transactions on Information Theory, 2007, 53(3): 1193-1196 https://doi.org/10.1109/TIT.2006.890727
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 https://doi.org/10.1016/S0166-218X(00)00350-4
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 https://doi.org/10.1016/j.ffa.2010.01.001
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 https://doi.org/10.1007/s00200-012-0182-8
28	Cao Y L, Gao J. Constructing quasi-cyclic codes from linear algebra theory. Designs, Codes and Cryptography, 2013, 67(1): 59-75 https://doi.org/10.1007/s10623-011-9586-x
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 https://doi.org/10.1109/18.681315

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