Perfect modes with distinct protective radii

We show that if the collection of all binary vectors of length n is partitioned into k spheres, then either k 2 or k n+2. Moreover, such partitions with k=n+2 are essentially unique. Academic PressRecently there has been some interest in the combinatorics of the geometry of the Hamming space, e.g., [10], and in particular, in tilings of this space [8]. Here, we investigate partitions of the Hamming space into spheres with possibly different radii. Such a partition is sometimes called a generalized perfect code, see e.g. [1,3,6,13,15]. Generalized spherepacking bounds can be found in [7]. Our aim in this note is to prove the following result (the gap-theorem): Theorem 1. If the binary Hamming space H(n, 2) is partitioned into M spheres, then article no. TA972816 388

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1997

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Perfect modes with distinct protective radii

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On partitions of the q‐ary Hamming space into few spheres

On partitions of the q‐ary Hamming space into few spheres

Tiling Hamming Space with Few Spheres

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