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

The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

A graph G has a (Formula presented.) -decomposition if there is a pair (Formula presented.) such that F is a subgraph of G and D is an acyclic orientation of (Formula presented.), where the maximum degree of F is no more than h and the maximum out-degree of D is no more than d. This paper proves that toroidal graphs having no adjacent triangles are (Formula presented.) -decomposable, and for (Formula presented.), toroidal graphs without i- and j-cycles are (Formula presented.) -decomposable. As consequences of these results, toroidal graphs without adjacent triangles are 1-defective DP-4-colorable, and toroidal graphs without i- and j-cycles are 1-defective DP-3-colorable for (Formula presented.).</p