几类最优五元负循环码的构造  

Construction of Several Classes of Optimal Quinary Negacyclic Codes

作　　者：杨锦 开晓山[1] YANG Jin;KAI Xiaoshan(School of Mathematics,Hefei University of Technology,Hefei 230601)

机构地区：[1]合肥工业大学数学学院,合肥230601

出　　处：《系统科学与数学》2024年第2期567-576,共10页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(61972126,12171134,12271137)资助课题。

摘　　要：作为循环码的推广,有限域上负循环码具有良好的代数结构.由于其具有高效的编码和译码算法,因而被广泛地应用在数据存储系统、通信系统和密码等领域.文章研究了码长n=(5^(m)-1)/2且具有两个零点βv和βv+2的五元负循环码,其中β是F5^(m)~*的生成元且0≤v≤(5^(m)-7)/2,通过分析有限域F5^(m)上方程组解的存在性,给出了这类码具有最优参数[(5^(m)-1)/2,(5^(m)-1)/2-2m,4]的充要条件.在此基础上,利用有限域F5^(m)上多项式唯一分解得到了两类最优五元负循环码.进一步,考虑了具有两个零点β^(v)和β^(v+2r)的五元负循环码,其中gcd(r,2n)=1,给出了这类五元负循环码具有极小距离4的充要条件,并构造了第三类最优五元负循环码.As a generalization of cyclic codes,negacyclic codes over finite fields have good algebraic structure.Due to efficient encoding and decoding algorithms,negacyclic codes over finite fields have many applications in various areas,such as data storage systems,communication systems and cryptography.In this paper,the authors investigate quinary negacyclic codes of length n=(5^(m)-1)/2 with two zerosβ"andβu+2,whereβis a generator of Fsm and 0≤u≤(5^(m)-7)/2.By analyzing the existence of solutions of some equations over F5^(m),necessary and sufficient conditions for such quinary negacyclic codes with optimal parameters[(5^(m)-1)/2,(5^(m)-1)/2-2m,4]are provided.On this basis,two new classes of optimal quinary negacyclic codes are constructed by using the unique factorizations of certain polynomials over F5^(m).Furthermore,the authors consider quinary negacyclic codes with two zeros Bu andβ^(u+2r),where gcd(r,2n)=1.Necessary and suficient conditions for such quinary negacyclic codes to have minimum distance four are provided and the third class of optimal quinary negacyclic codes are constructed.

关 键 词：负循环码 极小距离 极小多项式 分圆陪集 

分 类 号：O157.4[理学—数学]