Generalized multilevel constructions for binary Reed-Muller R(r, m) codes using projections onto GF (2q) are presented. These constructions exploit component codes over GF(2), GF(4), . . . ,GF(2q) that are based on shorter Reed-Muller codes, and set partitioning using partition chains of length-2l codes. This is then used for deriving multilevel constructions for the Barnes-Wall Λ(r, m) family of lattices. Similarly, the latter construction involves component codes over GF(2), GF(4),. . . ,GF(2q) and set partitioning based on partition chains of length-2l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.