Left dihedral codes over Galois rings GR(p2,m)

Let D2n=〈x,y∣xn=1,y2=1,yxy=x−1〉 be a dihedral group, and R=GR(p2,m) be a Galois ring of characteristic p2 and cardinality p2m where p is a prime. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let gcd(n,p)=1 in this paper. Then any left D2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes Ai and outer codes Ci, where Ai is a cyclic code over R of length n and Ci is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R. Specifically, a generator matrix and basic parameters for each outer code Ci are given. A formula to count the number of these codes is obtained and the dual code for each left D2n-code is determined. Moreover, all self-dual left D2n-codes and self-orthogonal left D2n-codes over R are presented.