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Balancing sets of vectors

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4 Author(s)
N. Alon ; Dept. of Math., Tel Aviv Univ., Israel ; E. E. Bergmann ; D. Coppersmith ; A. M. Odlyzko

For n>0, d⩾0, nd (mod 2), let K(n, d) denote the minimal cardinality of a family V of ±1 vectors of dimension n, such that for any ±1 vector w of dimension n there is a vV such that |v- w|⩽d, where v-w is the usual scalar product of v and w. A generalization of a simple construction due to D.E. Knuth (1986) shows that K(n , d)⩽[n/(d+1)]. A linear algebra proof is given here that this construction is optimal, so that K(n, d)-[n/(d+1)] for all nd (mod 2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links

Published in:

IEEE Transactions on Information Theory  (Volume:34 ,  Issue: 1 )