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=(5m-1)/2且具有两个零点βv和βv+2的五元负循环码,其中β是F5m的生成元且0 ≤ v ≤(5m-7)/2,通过分析有限域F5m上方程组解的存在性,给出了这类码具有最优参数[(5m-1)/2,(5m-1)/2-2m,4]的充要条件.在此基础上,利用有限域F5m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βv和βv+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.
Construction of Several Classes of Optimal Quinary Negacyclic Codes
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=(5m-1)/2 with two zerosβv and βv+2,where β is a generator of F*5m and 0 ≤ v ≤(5m-7)/2.By analyzing the existence of solutions of some equations over F5m,necessary and sufficient conditions for such quinary negacyclic codes with optimal parameters[(5m-1)/2,(5m-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 F5m.Furthermore,the authors consider quinary negacyclic codes with two zeros βv and βv+2r,where gcd(r,2n)=1.Necessary and sufficient conditions for such quinary negacyclic codes to have minimum distance four are provided and the third class of optimal quinary negacyclic codes are constructed.
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