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 Z(4), the composition of the automorphism group, and the structure of idempotents over Z(4). 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 Z(4) are shown to determine even unimodular lattices, including the extended quadratic residue code of length q + 1, where q = -1(mod 8) 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.