2016/09/14 by Cao, Yonglin, Cao, Yuan, Fu, Fang-Wei
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1609.04083
Let D2n=⟨ x,y| xn=1, y2=1, yxy=x-1⟩ be a dihedral group, and R=\rm 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 \rm 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 \cal Ai and outer codes Ci, where \cal 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 D2n-codes and self-orthogonal left D2n-codes over R are presented, respectively.