a b s t r a c tLet p be an odd prime. A family of (p − 1)-dimensional over-lattices yielding new record packings for several values of p in the interval [149 . . . 3001] is presented. The result is obtained by modifying Craig's construction and considering conveniently chosen Z-submodules of Q(ζ ), where ζ is a primitive pth root of unity. For p ≥ 59, it is shown that the center density of the (p − 1)-dimensional lattice in the new family is at least twice the center density of the (p − 1)-dimensional Craig lattice.

Let L/Q be a cyclic extension of degree p, where p is an odd unramified prime in L/Q. An explicit description of the integral trace form Tr L/Q (x 2)| O L , where O L is the ring of algebraic integers of L, is given, and an application to finding the minima of certain algebraic lattices is presented.

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A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n ≥ 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field ℚ(ζq) where q is the smallest prime congruent to 1 modulo n.

No abstract