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).