Reed-Muller codes: projections onto GF(4) and multilevel construction

A projection of binary Reed-Muller codes R(r,m) onto GF(4)^m-2 is presented. For an R(r,m) code, this operation yields a linear quaternary code with the same length, dimension, and minimum distance as the Reed-Muller R(r-1, m-2) code. Based upon this projection, multilevel construction is given for R(r,m), where the constituent codes applied to the different levels are themselves the Reed-Muller codes R(r-2, m-2) and R(r, m-2), as well as the aforementioned quaternary code. This construction of Reed-Muller codes is readily applicable for their efficient decoding