LCD codes and self-orthogonal codes in generalized dihedral group algebras (Q2205881)

A group code is a right ideal in a group ring \(R[G]\) where \(R\) is a commutative ring and \(G\) a finite group. In this paper, the authors consider a finite field \(\mathbb{F}_q\) and a generalized dihedral group \(D_{2n,r}\), \(\gcd(2n,q)=1\). As a first main result, they explicitly describe the primitive idempotents of the group algebra \(\mathbb{F}_q[D_{2n,r}]\). Given a code \(C\), it is named LCD whenever \(C \cap C^\perp = \{0\}\) and it is self-orthogonal iff \(C \subseteq C^\perp\), where \(C^\perp\) means dual of \(C\). LCD codes have cryptographic applications and self-orthogonal codes provide quantum codes. The second main result of the paper describes and counts LCD and self-orthogonal group codes in \(\mathbb{F}_q[D_{2n,r}]\).