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.