Quantum two-block group algebra codes

We consider quantum two-block group algebra (2BGA) codes, a family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle codes, except a cyclic group is replaced with an arbitrary finite group, generally non-Abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over Abelian groups, including quasicyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent binary connected 2BGA codes with stabilizer generator weights $W\le 8$, of length $n\le 100$ for Abelian groups, and $n\le 200$ for non-Abelian groups.