Number of nearest neighbors in a Euclidean code
K. Zeger, A. Gersho · IEEE Transactions on Information Theory · 1994
A Euclidean code is a finite set of points in n-dimensional Euclidean space /spl Rscr//sup n/. 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/sub n/ that can share the same nearest-neighbor codepoint is equal to the maximum kissing number /spl tau//sub 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 /spl tau//sub n/ to obtain F/sub n//spl les/2/sup n(0.401+o(1))/, which improves upon the best known upper known upper bound of F/sub n//spl les/2/sup n(1+o(1))/. We also show that the average touching number T of all the points in a Euclidean code is upper bounded /spl tau//sub n/.>