### Abstract

For n > 0, d≥ 0, n = d (mod2), 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 viv 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 Knuth 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 n = d (mod2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links.

Original language | English (US) |
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Pages (from-to) | 128-130 |

Number of pages | 3 |

Journal | IEEE Transactions on Information Theory |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Science Applications
- Library and Information Sciences

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## Cite this

Alon, N., Bergmann, E. E., Coppersmith, D., & Odlyzko, A. M. (1988). Balancing Sets of Vectors.

*IEEE Transactions on Information Theory*,*34*(1), 128-130. https://doi.org/10.1109/18.2610