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In this paper, new quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension, and the minimum distance, are studied. Using these constructions, new families of quaternary Reed-Muller (RM) codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. To make the duality relationships in the constructed families more evident, the concept of Kronecker inner product is introduced.