Quaternary quadratic residue codes and unimodular lattices

We construct new self-dual and isodual codes over the integers module 4. The binary images of these codes under the Gray map are nonlinear, but formally self-dual. The construction involves Hensel lifting of binary cyclic codes. Quaternary quadratic residue codes are obtained by Hensel lifting of the classical binary quadratic residue codes. Repeated Hensel lifting produces a universal code defined over the 2-adic integers. We investigate the connections between this universal code and the codes defined over Z₄, the composition of the automorphism group, and the structure of idempotents over Z₄. We also derive a square root bound on the minimum Lee weight, and explore the connections with the finite Fourier transform. Certain self-dual codes over Z₄ are shown to determine even unimodular lattices, including the extended quadratic residue code of length q+1, where q≡-1(mod8) is a prime power. When q=23, the quaternary Golay code determines the Leech lattice in this way. This is perhaps the simplest construction for this remarkable lattice that is known