Generalized Multilevel Constructions for Reed-Muller Codes and Barnes-Wall lattices

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