29 research outputs found
The weight distributions of a class of cyclic codes III
Recently, the weight distributions of the duals of the cyclic codes with two
zeros have been obtained for several cases. In this paper we solve one more
special case. The problem of finding the weight distribution is transformed
into a problem of evaluating certain character sums over finite fields, which
in turn can be solved by using the Jacobi sums directly
The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over F3
Cyclic codes have efficient encoding and decoding algorithms. The decoding
error probability and the undetected error probability are usually bounded by
or given from the weight distributions of the codes. Most researches are about
the determination of the weight distributions of cyclic codes with few
nonzeros, by using quadratic form and exponential sum but limited to low
moments. In this paper, we focus on the application of higher moments of the
exponential sum to determine the weight distributions of a class of ternary
cyclic codes with three nonzeros, combining with not only quadratic form but
also MacWilliams' identities. Another application of this paper is to emphasize
the computer algebra system Magma for the investigation of the higher moments.
In the end, the result is verified by one example using Matlab.Comment: 10 pages, 3 table
Weight distributions of cyclic codes with respect to pairwise coprime order elements
Let be an extension of a finite field with . Let
each be of order in and for .
We define a cyclic code over by
where
and . In this paper,
we present a method to compute the weights of . Further, we determine the weight distributions of the cyclic codes
and .Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1306.527
Weight distribution of two classes of cyclic codes with respect to two distinct order elements
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 be an extension of a finite field and ,
we determine the weight distribution of the cyclic codes 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 ; (2) \ord(g_1)=n,
, \ord(g_2)=\frac n 2, and