Self-dual binary 8m; 4m-codes constructed by left ideals of the dihedral group algebra 픽2[D8m]

Let $m$ be an arbitrary positive integer and $D_{8m}$ be the dihedral group of order $8m$ , i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^{2}=1, yxy=x^{-1}\rangle $ . Left ideals of the dihedral group algebra $\mathbb {F}_{2}[D_{8m}]$ are called binary left dihedral codes of length $8m$ , and abbreviated as binary left $D_{8m}$ -codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$ -codes. These codes make up an important class of self-dual binary $[8m,4m]$ -codes such that the dihedral group $D_{8m}$ is 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 $D_{8m}$ -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 $D_{48}$ -codes and left $D_{56}$ -codes respectively.