一类有限非链环上的循环码及其Gray映射  

A Family Cyclic codes of Finite Non-chain Rings and Gray map

作　　者：钟家伟 ZHONG Jiawei(Faculty of General Education and Foreign Languages,Anhui Institute of Information technology,Wuhu 241000)

机构地区：[1]安徽信息工程学院通识教育与外国语学院,芜湖241000

出　　处：《长春理工大学学报（自然科学版）》2022年第2期124-129,共6页Journal of Changchun University of Science and Technology(Natural Science Edition)

基　　金：安徽省高校自然科学重点项目(KJ2019A1286)。

摘　　要：有限非链环上的循环码是一类非常重要的线性码,近年来得到许多编码爱好者的广泛关注和研究,构造出了许多具有良好参数的最优码。有限非链环F_(2)+vF_(2)+uF_(2)+uvF_(2)(其中u2=0,v^(2)=v,uv=vu)上循环码在对抗量子通信中因消相干而产生噪声问题上具有良好的纠错功能。定义了有限非链环F_(2)+vF_(2)+uF_(2)+uvF_(2)到二元域上准循环码的Gray映射,并证明其为保距映射。研究得出环上循环码的生成多项式和幂等生成元的结构,通过所定义的Gray映射得出其上极小距离的计算方法。最后举例说明二元域上的准循环最优码的构造方法。Cyclic codes of Finite Non-chain Rings is a family of important linear codes,which inspire a widely concerned and study of code researchers recently,have been constructed many optimal codes with good parameters.Cyclic codes over finite non-chain rings F_(2)+vF_(2)+uF_(2)+uvF_(2)(u2=0,v^(2)=v,uv=vu)have good error-correcting function in counteracting noise caused by decoherence in quantum communication.In this paper,the Gray mapping of quasi-cyclic codes from finite non-chain rings to binary fields is defined,and prove its distance-preserving mapping.The structures of generating polynomials and idempotent generator of cyclic codes over a ring are obtained.By means of the defined Gray map,the computing method of the minimal distance on the Gray map is obtained.Finally,some example is given to illustrate the construction method of quasi-cyclic optimization on binary fields.

关 键 词：有限非链环 循环码 GRAY映射 准循环码 

分 类 号：TN911.22[电子电信—通信与信息系统]