Minimum weight bases for quaternary Reed – Muller codes

The quaternary Plotkin and BQ-Plotkin constructions giving the families of quaternary Reed – Muller codes were presented in 2009. The Gray map image of the obtained $\mathbb{Z}_4$-linear codes have the same parameters and fundamental properties as the codes in the classical binary linear Reed – Muller family. We have found one more general property for the families of quaternary Reed – Muller codes that is common with binary Reed – Muller codes: all these quaternary codes have bases of minimum weight codewords. The bases are constructed by induction.