Construction and enumeration of left dihedral codes satisfying certain duality properties.

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements and let $D_{2n}=\langle x,y\mid x^n=1, y^2=1, yxy=x^{n-1}\rangle$ be the dihedral group of order $n$. Left ideals of the group algebra $\mathbb{F}_{q}[D_{2n}]$ are known as left dihedral codes over $\mathbb{F}_{q}$ of length $2n$, and abbreviated as left $D_{2n}$-codes. Let ${\rm gcd}(n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over $\mathbb{F}_{q}$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over $\mathbb{F}_{q}$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over $\mathbb{F}_{q}$, and present several numerical examples to illustrative our applications.