Cyclic codes are an interesting type of linear codes and have wide
applications in communication and storage systems due to their efficient
encoding and decoding algorithms. Cyclic codes have been studied for many
years, but their weight distribution are known only for a few cases. In this
paper, let Frβ be an extension of a finite field Fqβ and r=qm,
we determine the weight distribution of the cyclic codes C={c(a,b):a,bβFrβ},c(a, b)=(\mbox {Tr}_{r/q}(ag_1^0+bg_2^0), \ldots, \mbox
{Tr}_{r/q}(ag_1^{n-1}+bg_2^{n-1})), g_1, g_2\in \Bbb F_r, in the following
two cases: (1) \ord(g_1)=n, n|r-1 and g2β=1; (2) \ord(g_1)=n,
g2β=g12β, \ord(g_2)=\frac n 2, m=2 and n2(rβ1)ββ£(q+1)
Let Frβ be an extension of a finite field Fqβ with r=qm. Let
each giβ be of order niβ in Frββ and gcd(niβ,njβ)=1 for 1β€iξ =jβ€u.
We define a cyclic code over Fqβ by
C(q,m,n1β,n2β,...,nuβ)β={c(a1β,a2β,...,auβ):a1β,a2β,...,auββFrβ}, where
c(a1β,a2β,...,auβ)=(Trr/qβ(i=1βuβaiβgi0β),...,Trr/qβ(i=1βuβaiβginβ1β)) and n=n1βn2β...nuβ. In this paper,
we present a method to compute the weights of C(q,m,n1β,n2β,...,nuβ)β. Further, we determine the weight distributions of the cyclic codes
C(q,m,n1β,n2β)β and C(q,m,n1β,n2β,1)β.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1306.527