On tilings of the binary vector space

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Abstract

Let n be a positive integer, L a subset of {0, 1,…,n}. We discuss the existence of partitions (or tilings) of the n-dimensional binary vector space Fn into L-spheres. By a L-sphere around an x in Fn we mean {y ϵ Fn, d(x, y) ϵ L}, d(x, y) being the Hamming distance betwe en x and y. These tilings are generalizations of perfect error correcting codes. We show that very few such tilings exist (Theorem 2) and characterize them all for any L ⊂ {0, 1,…,[12n]}.

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