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

Let D2n = 〈x, y | x = 1, y = 1, yxy = x〉 be a dihedral group, and R = GR(p, m) be a Galois ring of characteristic p and cardinality p 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 an extension Galois ring or principal ideal ring of R, and a generator matrix and basic parameters for each outer code Ci is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D2n-code is determined and all self-dual left D2ncodes and self-orthogonal left D2n-codes over R are presented, respectively.