2021/05/17 by Cao, Yuan, Cao, Yonglin, Ma, Fanghui · 1 citation
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2105.07924
Let \mathbbFq be the finite field of q elements and let D2n=⟨ x,y| xn=1, y2=1, yxy=xn-1⟩ be the dihedral group of order n. Left ideals of the group algebra \mathbbFq[D2n] are known as left dihedral codes over \mathbbFq of length 2n, and abbreviated as left D2n-codes. Let \rm gcd(n,q)=1. In this paper, we give an explicit representation for the Euclidean hull of every left D2n-code over \mathbbFq. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left D2n-codes over \mathbbFq. 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 \mathbbFq, and present several numerical examples to illustrative our applications.