Self-dual binary $[8m, 4m]$-codes constructed by left ideals of the dihedral group algebra $\mathbb{F}_2[D_{8m}]$

Let $m$ be an arbitrary positive integer and $D_{8m}$ be a 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 necessary 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.