Universal codes and unimodular lattices

Binary quadratic residue codes of length p + 1 produce via construction B and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction A modulo 4. We prove in a direct way the equivalence of these two constructions for p ~ 31. In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice BW32. The equivalence between construction B modulo 4 plus density doubling and construction A modulo 8 is also considered. In dimension 48 they both led to a new description of the extremal type II lattice P48q.