Let m be an arbitrary positive integer and D8mbe the dihedral group of order 8m, i.e., D8m= (x, y | x4m= 1, y2= 1, yxy = x-1). Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left D8m-codes. These codes make up an important class of self-dual binary [8m, 4m]-codes such that the dihedral group D8mis necessarily a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left D8m-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary [48, 24, 12]-code and an extremal self-dual binary [56, 28, 12]code from self-dual binary left D48-codes and left D56-codes respectively.

Let G(m,3,r)= (x, y | xm= 1, y3= 1, yx = xry) be a metacyclic group of order 3m, where gcd(m, r) = 1, 13≡ 1 (mod m). Then, left ideals of the group algebra Fq[G(m,3,r)] are called left metacyclic codes over Fq of length 3m, and abbreviated as left G(m,3,r)-codes. A system theory for left G(m,3,r)-codes is developed for the case of gcd(m, q) = 1 and r ≡ qE(mod m) for some positive integer ε, only using finite field theory and basic theory of cyclic codes and skew cyclic codes. The fact that any left G(m,3 1)-code is a direct sum of concatenated codes with inner codes λiand outer codes Ciis proved, where . Aiis a minimal cyclic code over Fqof length m and Ciis a skew cyclic code of length 3 over an extension field of Fq. Then, an explicit expression for each outer code in any concatenated code is provided. Moreover, the dual code of each left G(m,3,r)-code is given and self-orthogonal left G(m,3,r)-codes are determined.

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.