Quaternary quadratic residue codes and unimodular lattices

Abstract

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/sub 4/, the composition of the automorphism group, and the structure of idempotents over Z/sub 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/sub 4/ are shown to determine even unimodular lattices, including the extended quadratic residue code of length q+1, where q/spl equiv/-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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>