基于两类循环码构造3-设计  

Constructing 3-Design Based on Two Infinite Families of Cyclic Codes

作　　者：郑雪敏 唐春明[1] ZHENG Xuemin;TANG Chunming(College of Mathematics and Information,China West Normal Uiversity,Nanchong 637009,Sichuan)

机构地区：[1]西华师范大学数学与信息学院,四川南充637009

出　　处：《四川师范大学学报（自然科学版）》2023年第5期660-667,共8页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(11871058)。

摘　　要：信道编码理论中最热门的课题之一是利用组合设计和群论等数学知识构造新的循环码.由于循环码具有良好的代数结构,被广泛应用于工程和通信等领域.构造在F_(7^(m))上两类循环码族,第一类码的参数为[q+1,q-7,d],其中d≥6,m≥2且为整数;第二类码参数为[q+1,8,q-9],其中m≥2且为整数.设q=7^(m),由已给出的两类循环码的任意非零权重的码字的支撑集在一般射影线性群PGL(2,q)下是不变的,且一般射影线性群PGL(2,q)在射影直线PG(1,q)上的作用是3-传递的,从而可以验证对应的关联矩阵构造3-设计.One of the most popular topics in channel coding theory is the construction of new cyclic codes using mathematical knowledge such as combinatorial design and group theory.Due to the good algebraic structure of the cyclic code,it is widely used in engineering,communications and other fields.In this paper,two infinite families of cyclic codes are constructed on F_(7^(m)),and the parameters of the first type code are[q+1,q-7,d],where d≥6,m≥2 and m is an integer;the parameters of the second type code are[q+1,8,q-9],where m≥2 and m is an integer.Let q=7^(m),since the group action of the general projective linear group PGL(2,q)on the projective linearPG(1,q)is 3-transitive,the support set of all codewords for any fixed nonzero weight given by the two infinite families of cyclic codes is invariant under PGL(2,q),so that the corresponding incidence structure construction 3-design can be verified.

关 键 词：T-设计 线性码 一般射影线性群 

分 类 号：O157.4[理学—数学] O29[理学—基础数学]