Iterated squaring construction of bi-infinite group partition chain

Forney (1988) developed the concept of squaring construction. In particular, he derived code formulas for iterated squaring constructions of a finite group partition chain and applied them to construct the Reed-Muller codes and the Barnes-Wall lattices with its principal sublattices. We give a derivation for the code formula for the n times iterated squaring construction of a bi-infinite group partition chain. Our derivation is much simpler than the derivation given by Forney for the finite case. We also prove that the iterated squaring construction indeed generates the Reed-Muller codes. Finally, we derive formulas for the lattices Λ(r,n) and RΛ(r,n), which correct some of the errors of Forney