Hermitian duality of left dihedral codes over finite fields

Abstract

Let Fq2 be the finite field of q2 elements, where q is a power of a prime number, 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 Fq2[D2n] are known as left dihedral codes over Fq2 of length 2n, and abbreviated as left D2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit representation for the Hermitian dual code and the Hermitian hull of every left D2n-code over Fq2. On this basis, we determine all distinct Hermitian self-dual left D2n-codes, Hermitian linear complementary dual (LCD) left D2n-codes, and Hermitian self-orthogonal left D2n-codes over Fq2, respectively. Then we provide an explicit representation and a precise enumeration for these three subclasses of left D2n-codes. As an application, we provide several illustrative examples for obtaining Hermitian self-dual and Hermitian LCD left D2n-codes respectively.