On codes over R_k,m and constructions for new binary self-dual codes

In this work, we study codes over the ring R_k,m = F₂[u, v]/ u^k, v^m, uv - vu, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from R_{k, m} to F₂^km together with a Lee weight. After proving the MacWilliams identities for codes over R_k,m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over R_k,m. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].