NUMBER OF NEAREST NEIGHBORS IN A EUCLIDEAN CODE

A Euclidean code is a finite set of points in n-dimensional Euclidean space R(n). The total number of nearest neighbors of a given codepoint in the code is called its touching number. We show that the maximum n umber of codepoints F-n that can share the same nearest-neighbor codep oint is equal to the maximum kissing number tau(n) in n dimensions, th at is, the maximum number of unit spheres that can touch a given unit sphere without overlapping. We then apply a known upper bound on tau(n ) to obtain F-n less than or equal to 2(n(0.401+o(1))), which improves upon the best known upper bound of F-n less than or equal to 2(n(1+o( 1))). We also show that the average touching number T of all the point s in a Euclidean code is upper bounded by tau(n).