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 $p\le 31$. In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice $B{W}_{32}$. The equivalence between construction $B$ modulo $4$ plus density doubling and construction...