Linear codes over Z4+uZ4Z4+uZ4: MacWilliams identities, projections, and formally self-dual codes

Linear codes are considered over the ring Z4+uZ4Z4+uZ4, a non-chain extension of Z4Z4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z4+uZ4Z4+uZ4 to the rings Z4Z4 and F2+uF2F2+uF2 are considered and self-dual codes over Z4+uZ4Z4+uZ4 are studied in connection with these projections. A non-linear Gray map from Z4+uZ4Z4+uZ4 to (F2+uF2)2(F2+uF2)2 is defined together with real and complex lattices associated to codes over Z4+uZ4Z4+uZ4. Finally three constructions are given for formally self-dual codes over Z4+uZ4Z4+uZ4 and their Z4Z4-images together with some good examples of formally self-dual Z4Z4-codes obtained through these constructions.