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

Five Families of Three-Weight Ternary Cyclic Codes and Their Duals

As a subclass of linear codes, cyclic codes have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, five families of three-weight ternary cyclic codes whose duals have two zeros are presented. The weight distributions of the five families of cyclic codes are settled. The duals of two families of the cyclic codes are optimal

A Class of Three-Weight Cyclic Codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over \gf(p) whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page

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 $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ and $r=q^m$, we determine the weight distribution of the cyclic codes $\mathcal C=\{c(a, b): a, b \in \Bbb F_r\},$ 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 $g_2=1$; (2) \ord(g_1)=n, $g_2=g_1^2$, \ord(g_2)=\frac n 2, $m=2$ and $\frac{2(r-1)}n|(q+1)$

A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.Comment: 14 Page

Weight distributions of cyclic codes with respect to pairwise coprime order elements

Let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in $\Bbb F_r^*$ and $\gcd(n_i, n_j)=1$ for $1\leq i \neq j \leq u$. We define a cyclic code over $\Bbb F_q$ by $$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \in \Bbb F_r\},$$ where $$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\sum_{i=1}^ua_ig_i^{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight distributions of the cyclic codes $\mathcal C_{(q, m, n_1,n_2)}$ and $\mathcal C_{(q, m, n_1,n_2,1)}$.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1306.527