Barnes and Sloane recently described a "general construction" for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in U m with density A satisfying log2 A ~ — m logf m, as m -> oo, where logf m is the smallest value of k for which the fc-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96. §1. Introduction. In the past 23 years a series of papers (1-3 , 5, 8-12, 15-17) have described a variety of methods for packing equal spheres in Euclidean space. The present paper continues the sequence by simplifying the "general construction" of Barnes and Sloane (1), and thereby eliminating the requirement that the initial lattice be generated by its minimal vectors. This enables us to iterate the construction, obtaining lattice packings in U m with density A satisfying