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

Linear codes are considered over the ring ^Z4+u^Z4, a non-chain extension of ^Z4. 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+u^Z4 to the rings ^Z4 and ^F2+u^F2 are considered and self-dual codes over ^Z4+u^Z4 are studied in connection with these projections. A non-linear Gray map from ^Z4+u^Z4 to (^F2+u^F2)2 is defined together with real and complex lattices associated to codes over ^Z4+u^Z4. Finally three constructions are given for formally self-dual codes over ^Z4+u ^Z4 and their ^Z4-images together with some good examples of formally self-dual ^Z4-codes obtained through these constructions.