Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R = \mathbb{F}_{p^m}[u]/\langle u^2\rangle = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2 = 0)$, where $p$ is a prime and $m$ is a positive integer. For any $λ∈ \mathbb{F}_{p^m}^{×}$, an explicit representation for all distinct $λ$-constacyclic codes over $R$ of length $np^s$ is given by a canonical form decomposition for each code, where $s$ and $n$ are arbitrary positive integers satisfying ${\rm gcd}(p,n) = 1$. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length $np^s$ over $R$ are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over $\mathbb{F}_5+u\mathbb{F}_5$ of length $2· 3^t· 5^s$ are listed for any positive integer $t$.

In this paper we introduce the nonsubsampled shear let transform for multi-focus image fusion. In the proposed method, source images are decomposed by nonsubsampled shear let transform firstly. Then the decomposition coefficients are merged according to the given fusion rule. Finally the fused image is reconstructed by inverse nonsubsampled shear let transform. The experimental results over five pairs of registered multi-focus images and one pair of mis-registered multi-focus images demonstrate the superiority of the proposed method.

This paper is concerned with ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. These codes can be identified as submodules of the ring ${\mathbb{Z}}_{2}[x]/\langle x^r-1\rangle × {\mathbb{Z}}_{2}[x]/\langle x^s-1\rangle × {\mathbb{Z}}_{4}[x]/\langle x^t-1\rangle$. There are two major ingredients. First, we determine the generator polynomials and minimum generating sets of this kind of codes. Furthermore, we investigate their dual codes. We determine the structure of the dual of separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes completely. For the dual of non-separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes, we give their structural properties in a few special cases.