### Abstract

This paper studies the capacity of an ndimensional vector Gaussian noise channel subject to the constraint that an input must lie in the ball of radius R centered at the origin. It is known that in this setting the optimizing input distribution is supported on a finite number of concentric spheres. However, the number, the positions and the probabilities of the spheres are generally unknown. This paper characterizes necessary and sufficient conditions on the constraint R such that the input distribution supported on a single sphere is optimal. The maximum R ^{¯} n, such that using only a single sphere is optimal, is shown to be a solution of an integral equation. Moreover, it is shown that R ^{¯} n scales as √n and the exact limit of ^{√R} ^{¯n/} √n is found.

Original language | English (US) |
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Title of host publication | 2018 IEEE Information Theory Workshop, ITW 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

ISBN (Electronic) | 9781538635995 |

DOIs | |

State | Published - Jan 15 2019 |

Event | 2018 IEEE Information Theory Workshop, ITW 2018 - Guangzhou, China Duration: Nov 25 2018 → Nov 29 2018 |

### Publication series

Name | 2018 IEEE Information Theory Workshop, ITW 2018 |
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### Conference

Conference | 2018 IEEE Information Theory Workshop, ITW 2018 |
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Country | China |

City | Guangzhou |

Period | 11/25/18 → 11/29/18 |

### All Science Journal Classification (ASJC) codes

- Information Systems

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## Cite this

*2018 IEEE Information Theory Workshop, ITW 2018*[8613508] (2018 IEEE Information Theory Workshop, ITW 2018). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ITW.2018.8613508