最小距离为4的最优五元循环码

doi:10.11959/j.issn.1000-436x.2017030

Journal on Communications ›› 2017, Vol. 38 ›› Issue (2): 74-80.doi: 10.11959/j.issn.1000-436x.2017030

• Papers • Previous Articles Next Articles

Optimal quinary cyclic codes with minimum distance four

Ye TIAN¹,Yu-qing ZHANG^1,²,Yu-pu HU¹

¹ State Key Laboratory of Integrated Services Networks,Xidian University,Xi'an 710071,China
² National Computer Network Intrusion Protection Center,University of Chinese Academy of Sciences,Beijing 101408,China

Revised:2016-12-18 Online:2017-02-01 Published:2017-07-20
Supported by:
The National Key Research and Development Project(61572460);The National Key Research and Development Project(61272481);The National Natural Science Foundation of China(2016YFB0800703);The National Information Security Special Projects of National Development,the Reform Commission of China((2012)1424);China 111 Project(B16037)

Abstract

Abstract:

Cyclic codes are an extremely important subclass of linear codes.They are widely used in the communication systems and data storage systems because they have efficient encoding and decoding algorithm.Until now,how to construct the optimal ternary cyclic codes has received a lot of attention and much progress has been made.However,there is less research about the optimal quinary cyclic codes.Firstly,an efficient method to determine if cyclic codes C_(1,e,t)were optimal codes was obtained.Secondly,based on the proposed method,when the equation e=5_k+1 or e=5_m?2hold,the theorem that the cyclic codes C_(1,e,t)were optimal quinary cyclic codes was proved.In addition,perfect nonlinear monomials were used to construct optimal quinary cyclic codes with parameters[5_m?1,5_m?2m?2,4]optimal quinary cyclic codes over $F_{5^{m}}$ .

Key words: finite field, cyclic codes, minimum distance, perfect nonlinear function

CLC Number:

TN918

Ye TIAN,Yu-qing ZHANG,Yu-pu HU. Optimal quinary cyclic codes with minimum distance four[J]. Journal on Communications, 2017, 38(2): 74-80.

References 22

[1]	MACWILLIAMS F , SLOANE N . The theory of error-correcting codes[M]. North Holland: ElsevierPress, 1977.
[2]	ASSMUS E F , KEY J D . Designs and their codes[M]. Cambridge: Cambridge University PressPress, 1992.
[3]	MASSEY J L , . Some applications of coding theory in cryptography[C]// Fourth the Institute of Mathematics and its Applications (IMA) Conference on Cryptography and Coding. 1995: 33-47.
[4]	HUFFMAN W , PLESS V . Fundamentals of error-correcting codes[M]. Cambridge: Cambridge University PressPress, 2003.
[5]	DING C . Cyclic codes from the two-prime sequences[J]. IEEE Transactions on Information Theory, 2012,58(6): 3881-3891.
[6]	DING C . Cyclic codes from cyclotomic sequences of order four[J]. Finite Fields and Their Applications, 2013,23(96): 8-34.
[7]	CARLET C , DING C , YUAN J . Linear codes from highly nonlinear functions and their secret sharing schemes[J]. IEEE Transactions on Information Theory, 2005,51(6): 2089-2102.
[8]	DING C , HELLESETH T . Optimal ternary cyclic codes from monomials[J]. IEEE Transactions on Information Theory, 2013,59(9): 5898-5904.
[9]	LI N , ZHOU Z , HELLESETH T . On a conjecture about a class of optimal ternary cyclic codes[C]// Seventh International Workshop on Signal Design and its Applications in Communications. 2015: 62-65.
[10]	LI N , LI C , HELLESETH T ,et al. Optimal ternary cyclic codes with minimum distance four and five[J]. Finite Fields and Their Applications, 2014,30(6): 100-120.
[11]	LI C , LI N , HELLESETH T ,et al. The weight distributions of several classes of cyclic codes from APN monomials[J]. IEEE Transaction on Information Theory, 2014,60(8): 4710-4721.
[12]	XU G , CAO X , XU S . Optimal p-ary cyclic codes with minimum distance four from monomials[J]. Cryptography and Communications, 2016,8(4): 541-554.
[13]	YUAN J , CARLET C , DING C . The weight distribution of a class of linear codes from perfect nonlinear functions[J]. IEEE Transactions on Information Theory, 2006,52(2): 712-717.
[14]	ZHA Z , WANG X . Almost perfect nonlinear power functions in odd characteristic[J]. IEEE Transaction on Information Theory, 2011,57(7): 4826-4832.
[15]	FENG K , LUO J . Value distributions of exponential sums from perfect nonlinear functions and their applications[J]. IEEE Transactions on Information Theory, 2007,53(9): 3035-3041.
[16]	LI C , QU L , LING S . On the covering structures of two classes of linear codes from perfect nonlinear functions[J]. IEEE Transactions on Information Theory, 2009,55(1): 70-82.
[17]	VAN LINT J . A survey of perfect codes[J]. Journal of Mathematics, 1975,5(2): 199-224.
[18]	ZHOU Z , DING C . A class of three-weight cyclic codes[J]. Finite Fields and Their Applications, 2014,25: 79-93.
[19]	ZHOU Z , DING C . Seven classes of three-weight cyclic codes[J]. IEEE Transactions on Communications, 2013,61(10): 4120-4126.
[20]	YU L , LIU H . The weight distribution of a family of p-ary cyclic codes[J]. Designs,Codes and Cryptography, 2016,78(3): 1-15.
[21]	KAGEYAMA Y , MARUTA T . On the geometric constructions of optimal linear codes[J]. Designs,Codes and Cryptography, 2016,81(3): 469-480.
[22]	ZHENG D , WANG X , YU L . The weight enumerators of several classes of p-ary cyclic codes[J]. Discrete Mathematics, 2015,338(7): 1264-1276.

Metrics

Recommended 0

No Suggested Reading articles found!

Optimal quinary cyclic codes with minimum distance four

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

References 22

Related Articles 11

Metrics

Recommended 0

[1]	Ye TIAN,Yu-qing ZHANG,Yu-pu HU,Gao-fei WU. New bound of algebraic immunity of a class of Boolean function [J]. Journal on Communications, 2016, 37(10): 92-98.
[2]	Hua-wei HUANG,Chang-wen PENG,Yun-yun QU,Chun-hua LI. Security of the cryptosystems based on ergodic matrices [J]. Journal on Communications, 2015, 36(8): 61-67.
[3]	Ye-bin CHEN,Ying LI,Xiao ZHENG,Tao CHEN. On the average diameter of directed loop networks [J]. Journal on Communications, 2013, 34(2): 138-146.
[4]	Yong-zhe ZHAO,Bo ZHAO,Shi-hui PEI,Zhan-hua JIANG. Design and implement on the HFEM public key scheme [J]. Journal on Communications, 2011, 32(6): 24-31.
[5]	Xiu-sheng LIU. Cyclic codes over formal power series rings [J]. Journal on Communications, 2011, 32(2): 68-71.
[6]	Zhong LI,Yi WANG,Dai-yuan PENG. Sliding window multiplication in finite field GF(2ⁿ) [J]. Journal on Communications, 2008, 29(7): 27-31.
[7]	Yong DING,Feng GUI. High speed modular divider based on GCD algorithm over GF（2^m） [J]. Journal on Communications, 2008, 29(10): 199-204.
[8]	Pin-hui KE,Jie ZHANG,Qiao-yan WEN. Further study of the trace representation of Bent sequences families [J]. Journal on Communications, 2007, 28(5): 118-121.
[9]	Shan-bao HE,Chun-ming ZHAO,Zhi-hua SHI,Ming JIANG. Low-density parity-check codes based on sparse binary sequences [J]. Journal on Communications, 2005, 26(6): 81-86.
[10]	Qing-po ZHANG,Cai-yun CHEN,Lu-sheng CHEN,Yan-ling CHEN. ElGamal cryptosystem and digital signature scheme based on polynomials over finite fields [J]. Journal on Communications, 2005, 26(5): 69-72.
[11]	Pin-hui KE,Zu-ling CHANG,Qiao-yan WEN. Construction of generalized binary bent sequences [J]. Journal on Communications, 2005, 26(12): 19-23.