CODES AND LATTICES

Abstract

One of the most important questions in the theory of N-dimensional Euclidean lattices is: How many minima can be found in an N-Iattice? As first result G. F. Voronoi proved in [1] that this number is not greater than 2N +1 - 2. On the other hand, for the well-known classical extremal lattices, this number is not 'enough large', in these lattices there are only O(N2) minimal vectors. The first lattices with a lot of minima were constructed by E. S. Barnes and G. E. Wall. They proved in [1] that in the dimensions N = 2n there exists such an N-Iattice in which the number of minima is SUN) = C, (N~(log2 N+l») = c . (2~[(log2N)2+1og2Nl). (The assymptotic formula was given by J.Leech in [3].) The above mentioned lattice for dimension N = 23 is the well-known lattice Eg. Using the base properties of the Reed-Muller code, in this paper we give the characterization of the minima of this lattice and determine the number of minima of the 2n - 2-dimensional lattice that is a generalization of the extremal lattice Es. We note that the author proved some similar results in the paper [4] but the precise value of the above number was not known yet.