Projective Linear Codes Based on the Quadratic Multiplicative Characters

CHEN Fu-ling; HENG Zi-ling; WANG Xin-ran; LI Cheng-ju

doi:10.12263/DZXB.20211633

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Projective Linear Codes Based on the Quadratic Multiplicative Characters

PAPERS | 更新时间：2025-07-02

- Projective Linear Codes Based on the Quadratic Multiplicative Characters
- ACTA ELECTRONICA SINICA Vol. 51, Issue 1, Pages: 32-41(2023)
- 作者机构：
  
  1.长安大学理学院,陕西西安 710064
  2.东南大学移动通信国家重点实验室,江苏南京 210096
  3.华东师范大学高可信计算重点实验室,上海 200062
- 作者简介：
- 基金信息：
  
  Foundation Item(s):　National Natural Science Foundation of China(11901049;12071138);The Young Talent Fund of University Association for Science and Technology in Shaanxi, China(20200505);The Open Research Fund of National Mobile Communications Research Laboratory, Southeast University(2022D05);The Fundamental Research Funds for the Central Universities, CHD(300102122202);Shanghai Trusted Industry Internet Software Collaborative Innovation Center
- DOI：10.12263/DZXB.20211633
  CLC： O236.2
- Received：08 December 2021，
  
  Revised：2022-05-15，
  
  Published：25 January 2023
- 稿件说明：
移动端阅览
陈辅灵,衡子灵,王鑫然等.基于二次乘法特征的射影线性码[J].电子学报,2023,51(01):32-41.

CHEN Fu-ling,HENG Zi-ling,WANG Xin-ran,et al.Projective Linear Codes Based on the Quadratic Multiplicative Characters[J].ACTA ELECTRONICA SINICA,2023,51(01):32-41.
陈辅灵,衡子灵,王鑫然等.基于二次乘法特征的射影线性码[J].电子学报,2023,51(01):32-41. DOI： 10.12263/DZXB.20211633.

CHEN Fu-ling,HENG Zi-ling,WANG Xin-ran,et al.Projective Linear Codes Based on the Quadratic Multiplicative Characters[J].ACTA ELECTRONICA SINICA,2023,51(01):32-41. DOI： 10.12263/DZXB.20211633.

摘要

基于有限域上的二次乘法特征构造了两类线性码，精确计算出了它们的参数和重量分布.结果表明，第一类线性码是射影三重码，且对偶码关于球填充界几乎最优；第二类线性码是射影二重码，且对偶码关于球填充界几乎最优.此外，本文还得到了一些自正交码和极小码，它们可分别用于构造量子码和安全高效访问结构上的密钥共享方案.

Abstract

Two families of linear codes are constructed based on the quadratic multiplicative characters of finite fields. The parameters and weight distributions of the codes are explicitly determined. It turns out that the first family of linear codes are projective three-weight ones whose duals are almost optimal according to the sphere-packing bound. The second family of linear codes are projective two-weight ones whose duals are also almost optimal according to the sphere-packing bound. Besides

some self-orthogonal codes and minimal codes are obtained. The self-orthogonal codes can be used to construct quantum codes and minimal codes can be used to construct secret sharing schemes with safe and sufficient access structures.

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references

冯克勤 , 刘凤梅 . 代数与通信 [M]. 北京 : 高等教育出版社 , 2005 .

GRIESMER J H . A bound for error-correcting codes [J]. IBM Journal of Research and Development , 1960 , 4 : 532 - 542 .

TANG Chun-ming , LI Nian , QI Yan-feng , ZHOU Zheng-chun , HELLESETH Tor . Linear codes with two or three weights from weakly regular bent functions [J]. IEEE Transactions on Information Theory , 2016 , 62 ( 3 ): 1166 - 1176 .

HENG Zi-ling , LI De-xiang , DU Jiao , CHEN Fu-ling . A family of projective two-weight linear codes [J]. Designs, Codes and Cryptography , 2021 , 89 : 1993 - 2007 .

YANG Shu-di , YAO Zheng-an , ZHAO Chang-an . A class of three-weight linear codes and their complete weight enumerators [J]. Cryptography and Communications , 2017 , 9 : 133 - 149 .

YANG Shu-di , YAO Zheng-an . Complete weight enumerators of a family of three-weight linear codes [J]. Designs, Codes and Cryptography , 2017 , 82 : 663 - 674 .

YANG Shu-di , YAO Zheng-an , ZHAO Chang-an . The weight distributions of two classes of p ary cyclic codes with few weights [J]. Finite Fields and Their Applications , 2017 , 44 : 76 - 91 .

杨淑娣 , 唐春明 . 循环码的完全重量分布 [J]. 江苏师范大学学报(自然科学版) , 2018 , 36 ( 2 ): 64 - 68 .

YANG Shu-di , TANG Chun-ming . The complete weight enumerator of cyclic codes [J]. Journal of Jiangsu Normal University , 2018 , 36 ( 2 ): 64 - 68 . (in Chinese)

杨淑娣 , 岳勤 . 一类线性码的完全重量分布 [J]. 计算机工程与科学 , 2019 , 41 ( 2 ): 281 - 285 .

YANG Shu-di , YUE Qin . Complete weight enumerators of a class of linear codes [J]. Computer Engineering and Science , 2019 , 41 ( 2 ): 281 - 285 . (in Chinese)

HENG Zi-ling , WANG Qiu-yan , DING Cun-sheng . Two families of optimal linear codes and their subfield codes [J]. IEEE Transactions on Information Theory , 2020 , 66 ( 11 ): 6872 - 6883 .

HENG Zi-ling , DING Cun-sheng , WANG Wei-qiong . Optimal binary linear codes from maximal arcs [J]. IEEE Transactions on Information Theory , 2020 , 66 ( 9 ): 5387 - 5394 .

杜小妮 , 吕红霞 , 王蓉 , 李丽 . 两类四重线性码的构造 [J]. 西北师范大学学报(自然科学版) , 2018 , 54 ( 6 ): 1 - 4 .

DU Xiao-ni , LV Hong-xia , WANG Rong , LI Li . A construction of two classes of linear codes with four-weights [J]. Journal of Northwestern Normal University (Natural Science Edition) , 2018 , 54 ( 6 ): 1 - 4 . (in Chinese)

杜小妮 , 吕红霞 , 王蓉 . 一类四重和六重线性码的构造 [J]. 电子与信息学报 , 2019 , 41 ( 12 ): 2995 - 2999 .

DU Xiao-ni , LV Hong-xia , Wang Rong . Construction of a class of linear codes with four-weight and six-weight [J]. Journal of Electronics and Information Technology , 2019 , 41 ( 12 ): 2995 - 2999 . (in Chinese)

DU Xiao-ni , WANG Rong , TANG Chun-ming , WANG Qi . Infinite families of 2-designs from linear codes [J]. Applicable Algebra in Engineering Communication and Computing , 2022 , 33 : 193 - 211 .

DU Xiao-ni , LI Xiao-dan , WAN Yun-qi . A class of linear codes with three and five weights [J]. Chinese Journal of Electronics , 2019 , 28 ( 03 ): 457 - 461 .

胡丽琴 , 岳勤 , 朱小萌 . 具有两个非零点循环码的权重分布 [J]. 中国科学: 数学 , 2014 , 44 ( 9 ): 1021 - 1034 .

HU Li-qin , YUE Qin , ZHU Xiao-meng . Weight distributions of a class of cyclic codes with two non-zeros [J]. Science in China: Mathematics , 2014 , 44 ( 9 ): 1021 - 1034 . (in Chinese)

FENG Tao . On cyclic codes of length 2 2r -1 with two zeros whose dual codes have three weights [J]. Designs, Codes and Cryptography , 2012 , 62 ( 3 ): 253 - 258 .

XIANG Can , WANG Xian-Fang , TANG Chun-ming , FU Fang-wei . Two classes of linear codes and their weight distributions [J]. Applicable Algebra in Engineering Communication and Computing , 2018 , 29 ( 3 ): 209 - 225 .

ZHOU Zheng-chun , LI Nian , FAN Cui-ling , HELLESETH Tor . Linear codes with two or three weights from quadratic Bent functions [J]. Designs, Codes and Cryptography , 2016 , 81 ( 2 ): 283 - 295 .

DING Cun-sheng , NIEDERREITER H . Cyclotomic linear codes of order 3 [J]. IEEE Transactions on Information Theory , 2007 , 53 ( 6 ): 2274 - 2277 .

DING Ke-lan , DING Cun-sheng . A class of two-weight and three-weight codes and their applications in secret sharing [J]. IEEE Transactions on Information Theory , 2015 , 61 : 5835 - 5842 .

LI Cheng-ju , BAE S , AHN J , Yang Shu-di , YAO Zheng-an . Complete weight enumerators of some linear codes and their applications [J]. Designs Codes Cryptography , 2016 , 81 : 153 - 168 .

高健 , 吕京杰 . <math id="M480"><msub><mrow><mi mathvariant="bold">ℤ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>u</mi><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=41699905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=41699901&type= 18.45733452 3.89466691 上的一类循环码 [J]. 电子学报 , 2018 , 46 ( 7 ): 1768 - 1773 .

GAO Jian , LÜ Jing-jie . A class of cyclic codes on <math id="M481"><msub><mrow><mi mathvariant="bold">ℤ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>u</mi><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=41699905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=41699901&type= 18.45733452 3.89466691 [J]. Acta Electronica Sinica , 2018 , 46 ( 7 ): 1768 - 1773 . (in Chinese)

HUFFMAN W C , PLESS V . Fundamentals of Error-Correcting Codes [M]. Cambridge : Cambridge University Press , 2003 .

LIDL R , NIEDERREITER H . Finite Fields [M]. Boston : Addison-Wesley , 1983 .

KETKAR A , KLAPPENECKER A , KUMAR S . Nonbinary stabilizer codes over finite fields [J]. IEEE Trans Inf Theory , 2006 , 52 : 4892 - 4914 .

ASHIKHMIN A , BARG A . Minimal vectors in linear codes [J]. IEEE Trans Inf Theory , 1998 , 44 : 2010 - 2017 .

CALDERBANK R , KANTER W M . The geometry of two-weight codes [J]. Bull Lond Math Soc , 1986 , 18 : 97 - 122 .

CALDERBANK R , GOETHAL J M . Three-weight codes and association schemes [J]. Philips J Res , 1984 , 39 : 143 - 152 .

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