\(G\)-codes, self-dual \(G\)-codes and reversible \(G\)-codes over the ring \(\mathscr{B}_{j,k} \) (Q2120996)

In this paper, the authors study a new family of rings, denoted by \(\mathcal{B}_{j,k}\) with base field \(\mathbb{F}_{p^r}\) for prime \(p\) and positive integer \(r\). After introducing the arithmetics of these rings in Section 2, the authors define a Gray map in Section 2.1 and they prove that the Gray map of a linear code over \(\mathcal{B}_{j,k}\) is a linear code over \(\mathbb{F}_{p^r}.\) (Lemma 2.10). Further, the authors investigate the automorphism group of these codes and they find that some swap maps are always contained in their automorphism group if the base field has characteristic \(2\) (Theorem 2.11). The dual code of the Gray map can be computed as the Gray map of the dual code and therefore the Gray map of a self-dual code is also a Gray map (Theorem 2.13 and Corollary 2.14). In Section 2.2, the authors show that \(\mathcal{B}_{j,k}\) is a Frobenius ring and they give its generating character (Theorem 2.16). In Section 3, the authors introduce \(G\)-codes over \(\mathcal{B}_{j,k}\). The authors prove that the automorphism group of a Gray image of a linear \(G\)-code necessarily contains \(G\) as a subgroup. (Theorem 3.5) In Section 4, the authors investigate projections and lifts of self-dual \(G\)-codes. Lifts of self-dual codes are self-orthogonal codes over another ring from the same family. Finally, in Section 5 and 6, the authors prove some statements about self-dual \(G\)-codes and reversible codes.