A Simple Construction for the Barnes-Wall Lattices by Gabriele Nebe Abteilung Reine Mathematik Universitaet Ulm, 89069 Ulm, Germany and E. M. Rains and N. J. A. Sloane Information Sciences Research, AT&T Shannon Labs 180 Park Avenue, Florham Park, NJ 07932-0971, U.S.A. To Dave Forney, on the occasion of his sixtieth birthday ABSTRACT A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized Barnes-Wall lattices. This leads to an especially simple construction for the usual Barnes-Wall lattices. This is based on the third author's talk at the Forney-Fest, M.I.T., March 2000, which in turn is based on our paper "The Invariants of the Clifford Groups", Designs, Codes, Crypt., 24 (2001), 99--121, to which the reader is referred for further details and proofs. For the full version of the present paper, see http://NeilSloane.com/doc/bw.pdf (pdf) or http://NeilSloane.com/doc/bw.ps (ps).