Because the matroid search algorithm is very complicated and the local matroid search algorithm can not search all optimal codes, this paper proposes a variable matroid search algorithm to search the quasi-cyclic codes by researching matroid search algorithm. The algorithm reduces the computational complexity by reducing the repeated search. Based on this algorithm, the systematic binary quasi-cyclic codes of which the rate is 1/p are constructed. With the change of integer p, the optimal codes of rate 1/p can be obtained by the generator matrix reducing or adding a loop matrix. Through experiments, two new codes of which the minimum distance is larger than the existing optimal codes are worked out, which indicate the feasibility and superiority of the algorithm.
TOWNSEND R and WELDON E. Self-orthogonal quasi-cyclic codes[J]. IEEE Transactions on Information Theory, 1967, 13(2): 183-195. doi: 10.1109/TIT.1967. 1053974.
[2]
CHEN Z. New results on binary quasi-cyclic codes[C]. Proceedings of IEEE International Symposium on Information Theory, Sorrento Italy, 2000: 151-154.
[3]
HEIJNEN P, VAN T H, VERHOEFF T, et al. Some new binary quasi-cyclic codes[J]. IEEE Transactions on Information Theory, 1998, 44(5): 1994-1996. doi: 10.1109/ 18.705580.
ZHANG Yi, DA Xinyu, and SU Yidong. Construction of quasi-cyclic low-density parity-check codes with a large girth based on arithmetic progression[J]. Journal of Electronics & Information Technology, 2015, 37(2): 394-398. doi: 10.11999/JEIT140538．
[5]
CHEN Z. Database of quasi-twisted codes[OL]. http:// moodle. tec.hkr.se/~chen/research/codes, 2015.9.
GUO Rui, LIU Chunyu, ZHANG Hua, et al. Full diversity LDPC codes design and energy efficiency analysis for clustering wireless sensor networks[J]. Journal of Electronics & Information Technology, 2015, 37(7): 1580-1585. doi: 10.11999/JEIT141294.
CHEN Zhenhua, XU Xiaomei, CHEN Yougan, et al. Design and analysis of Protograph-based LDPC codes in shallow water acoustic channels[J]. Journal of Electronics & Information Technology, 2016, 38(1): 153-159. doi: 10.11999/ JEIT150415.
LAN Yazhu, YANG Haigang, and LIN Yu. Design of dynamic adaptive LDPC decoder based on FPGA[J]. Journal of Electronics & Information Technology, 2015, 37(8): 1937-1943. doi: 10.11999/JEIT141609.
[9]
WHITNEY H. On the abstract properties of linear dependence[J]. The American Mathematical Society, 1935, 57(1): 509-533. doi: 10.1007/978-1-4612-2972-8_10.
[10]
GREENE C. Weight enumeration and the geometry of linear codes[J]. Studies in Applied Mathematics, 1976, 55(55): 119-128. doi: 10.1002/sapm1976552119.
[11]
BARG A. The matroid of supports of a linear code[J]. Applicable Algebra in Engineering Communication & Computing, 1997, 8(2): 165-172. doi: 10.1007/s 002000050060.
[12]
KASHYAP N. A decomposition theory for binary linear codes[J]. IEEE Transactions on Information Theory, 2008, 54(7): 3035-3058. doi: 10.1109/TIT.2008.924700.
WU Guangfu and WANG Lin. Design of a short high rate LDPC code[J]. Journal of Applied Sciences, 2013, 31(6): 559-563. doi: 10.3969/j.issn.0255-8297.2013.06.002.
[14]
WU Guangfu, WANG Lin, and TRUONG T K. Use of matroid theory to construct a class of good binary linear codes[J]. IET Communications, 2014, 8(6): 893-898. doi: 10.1049/iet-com.2013.0671.
[15]
WU Guangfu, Chang H C, WANG Lin, et al. Constructing rate 1/p systematic binary quasi-cyclic codes based on the matroid theory[J]. Designs Codes and Cryptography, 2014, 71(1): 47-56. doi: 10.1007/s10623-012-9715-1.
[16]
WU Guangfu, LI Yong, ZHANG Shuiping, et al. A random local matroid search algorithm to construct good rate 1/p systematic binary Quasi-Cyclic codes[J]. IEEE Communications Letters, 2015, 19(5): 699-702. doi: 10.1109/ LCOMM.2015.2401572.
[17]
OXLEY J. Matroid Theory [M]. Oxford U K, Oxford University Press, 2011: 5-26．
[18]
TILBURG H C A V. On quasi-cyclic codes with rate 1/m [J]. IEEE Transactions on Information Theory, 1978, 24(5): 628-629. doi: 10.1109/TIT.1978.1055929.
[19]
GULLIVER T A and BHARGAVA V K. An updated table of rate 1/p binary quasi-cyclic codes[J]. Applied Mathematics Letters, 1995, 8(5): 81-86. doi: 10.1016/0893-9659 (95)00071-W.
[20]
GRASSL M. Tables of linear codes and quantum codes [OL]. http://www.codetables.de, 2015.9.
