Left dihedral codes over Galois rings ${\rm GR}(p^2,m)$

Let $D_{2n}=\langle x,y\mid x^n=1, y^2=1, yxy=x^{-1}\rangle$ be a dihedral group, and $R={\rm GR}(p^2,m)$ be a Galois ring of characteristic $p^2$ and cardinality $p^{2m}$ where $p$ is a prime. Left ideals of the group ring $R[D_{2n}]$ are called left dihedral codes over $R$ of length $2n$, and abbreviated as left $D_{2n}$-codes over $R$. Let ${\rm gcd}(n,p)=1$ in this paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a direct sum of concatenated codes with inner codes ${\cal A}_i$ and outer codes $C_i$, where ${\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$ is a skew cyclic code of length $2$ over an extension Galois ring or principal ideal ring of $R$, and a generator matrix and basic parameters for each outer code $C_i$ is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left $D_{2n}$-code is determined and all self-dual left $D_{2n}$-codes and self-orthogonal left $D_{2n}$-codes over $R$ are presented, respectively.