Abstract
Let D2n be the dihedral group of order n. Left ideals of the group algebra FqD2n are known as left dihedral codes over Fq of length 2n, and abbreviated as left D2n-codes. In this paper, a system theory for left D2n-codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left D2n-code is a direct sum of concatenated codes with inner codes Ai and outer codes Ci, where Ai is a minimal self-reciprocal cyclic code over Fq of length n and Ci is a skew cyclic code of length 2 over an extension field or principal ideal ring of Fq. Then for the case of gcd(n,q)=1, we give a precise description for outer codes in the concatenated codes, provide the dual code for any left D2n-code and determine all self-dual left D2n-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 D2pn-codes over Fq is investigated.
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
