Number of nearest neighbors in a Euclidean code

A Euclidean code is a finite set of points in n-dimensional Euclidean space ℛⁿ. The total number of nearest neighbors of a given codepoint in the code is called its touching number. We show that the maximum number of codepoints F_n that can share the same nearest-neighbor codepoint is equal to the maximum kissing number τ_n in n dimensions, that is, the maximum number of unit spheres that can touch a given unit sphere without overlapping. We then apply a known upper bound on τ_n to obtain F_n⩽2^{n(0.401+o(1))}, which improves upon the best known upper known upper bound of F_n⩽2^n(1+o(1)). We also show that the average touching number T of all the points in a Euclidean code is upper bounded τ_n