Frames in the odd Leech lattice

In this paper, we show that there is a frame of norm k in the odd Leech lattice for every k\ge 3.Comment: 7 page

On a property of 2-dimensional integral Euclidean lattices

Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through exactly $n$ points of $L$.Comment: 9 page

On Euclidean designs and the potential energy

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, We formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean a-code if any distinct two points in the set are separated at least by a. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean a-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean t-designs as an analogue of the linear programming bound.Comment: 14 page