Concatenated structure of left dihedral codes

• We develop a system theory for left dihedral codes only using finite field theory, basic theory of cyclic codes and skew cyclic codes.
• We prove that any left dihedral code is a direct sum of concatenated codes in which the inner codes and outer codes are cyclic codes and skew cyclic codes respectively.
• We provide a precise expression for all distinct left D2nD2n-code over FqFq, where D2nD2n is the dihedral group of order n   and gcd⁡(n,q)=1gcd⁡(n,q)=1.
• We give the dual code of any left D2nD2n-code and list all self-dual left D2nD2n-codes explicitly.

Let D2nD2n be the dihedral group of order n  . Left ideals of the group algebra FqD2nFqD2n are known as left dihedral codes over FqFq of length 2n  , and abbreviated as left D2nD2n-codes. In this paper, a system theory for left D2nD2n-codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left D2nD2n-code is a direct sum of concatenated codes with inner codes AiAi and outer codes CiCi, where AiAi is a minimal self-reciprocal cyclic code over FqFq of length n   and CiCi is a skew cyclic code of length 2 over an extension field or principal ideal ring of FqFq. Then for the case of gcd⁡(n,q)=1gcd⁡(n,q)=1, we give a precise description for outer codes in the concatenated codes, provide the dual code for any left D2nD2n-code and determine all self-dual left D2nD2n-codes. Moreover, all 1995 binary left dihedral codes and all 255 binary self-dual left dihedral codes of length 30 are given, and a class of left D2pnD2pn-codes over FqFq is investigated.