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

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