具有两个非零点循环码的权重分布

循环码作为一类重要的线性码,因其有效的编码和译码算法而被广泛应用于通信和存储系统.令F_r为有限域F_q的一个扩域,其中r=q~m,α为有限域F_r的本原元.设n=n1n2满足gcd(n1,n2)=1为r-1的因子.定义F_q上的一类循环码C={c(a1,a2)=(T_r/q(a1y_1~2+a2(g1g2)~i))_(i=0)~(n-1):a1,a2∈F_r},其中g1=α(r-1)/(n1),g2=α(r-1)/(n2),且g1与g1g2不共轭.本文将利用Gauss周期刻画循环码C的权重分布.特别地,这类循环码包含一类二重循环码和一类三重循环码.

Weight distributions of a class of cyclic codes with two nonzeros

Cyclic codes are an important type of linear codes and have wide applications in communication and storage systems because of their efficient encoding and decoding algorithms.In this paper,let Fr be an extension of a finite field Fq with r = qm,and α be a generator of Fr＊.Let n = n1n2be a positive divisor of r- 1 such that gcd（n1,n2） = 1.We define a class of cyclic codes over Fqby C={c（a1,a2）=（Tr/q（a1gi1＋a2（g1g2）i）i=0n-1：a1,a2∈Fr},where g1=αr-1/n1,g2=αr-1/n2,and g_1 and g1g2 are not conjugate.In this paper,we determine the weight distribution of the cyclic codes C using Gauss periods.As applications,we obtain a class of 2-weight cyclic codes and a class of 3-weight cyclic codes.