The well known Plotkin construction is, in the current paper, generalized and used to yield new families of ℤ
-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of ℤ
-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
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