On a refinement of Craig’s lattices

Abstract

Let 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 p th 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.