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Number of nearest neighbors in a Euclidean code

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2 Author(s)
Zeger, K. ; Dept. of Electr. Eng., Illinois Univ., Urbana, IL, USA ; Gersho, A.

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 Fn that can share the same nearest-neighbor codepoint is equal to the maximum kissing number τn 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 τn to obtain Fn⩽2n(0.401+o(1)), which improves upon the best known upper known upper bound of Fn⩽2n(1+o(1)). We also show that the average touching number T of all the points in a Euclidean code is upper bounded τn

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Information Theory, IEEE Transactions on  (Volume:40 ,  Issue: 5 )