Construction and enumeration of left dihedral codes satisfying certain duality properties

Abstract

Let Fq be the finite field of q elements and let D2n=〈x,y|xn=1,y2=1,yxy=xn−1〉 be the dihedral group of 2n elements. Left ideals of the group algebra Fq[D2n] are known as left dihedral codes over Fq of length 2n, and abbreviated as left D2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit representation for the Euclidean hull of every left D2n-code over Fq. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left D2n-codes over Fq. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left D2n-codes and self-dual left D2n-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left D2n-code over Fq, and present several numerical examples to illustrative our applications.