### Abstract

This paper discusses a heuristic solution procedure for a combinatorial optimization problem that originates in designing signal constellations for modems. The design problem is to place m signals in a two‐dimensional space to minimize the average error rate under specified noise conditions, using a maximum‐likelihood decoding scheme. Intuitively, it amounts (roughly) to spreading the signal points as far apart as possible, according to the distance measurement implied by the noise function. We show how this problem can be reduced to a discrete one: Given an ℓ by n matrix P, and m < ℓ, find an m‐row subset M = {i_{1}, ···, i_{m}} of the rows of P that maximizes (Formula Presented.) and then describe an efficient procedure for finding this maximizing set. Experiments indicate that the procedure is a useful tool, both for analysis of existing and proposed signal constellations and for finding new, near‐optimum ones.

Original language | English (US) |
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Pages (from-to) | 1145-1159 |

Number of pages | 15 |

Journal | Bell System Technical Journal |

Volume | 52 |

Issue number | 7 |

DOIs | |

State | Published - Sep 1973 |

### All Science Journal Classification (ASJC) codes

- Engineering(all)

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## Cite this

*Bell System Technical Journal*,

*52*(7), 1145-1159. https://doi.org/10.1002/j.1538-7305.1973.tb02009.x