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Recently, new families of quaternary linear Reed-Muller codes have been introduced. They satisfy that, after the Gray map, the corresponding BBZ4-linear codes have the same parameters and properties as the codes of the binary linear Reed-Muller family. A structural invariant, the dimension of the kernel, for binary codes is used to classify completely these BBZ4-linear codes. The dimension of the kernel for these BBZ4 -linear codes is established generalizing the known results about the dimension of the kernel for BBZ4 -linear Hadamard and BBZ4 -linear extended 1-perfect codes.