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The Optimal Lattice Quantizer in Nine Dimensions

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Bruce Allen
  • Erik Agrell

Research Organisations

External Research Organisations

  • Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
  • Chalmers University of Technology

Details

Original languageEnglish
Article number2100259
JournalAnnalen der Physik
Volume533
Issue number12
Early online date24 Oct 2021
Publication statusPublished - 14 Dec 2021

Abstract

The optimal lattice quantizer is the lattice that minimizes the (dimensionless) second moment G. In dimensions 1 to 3, it has been proven that the optimal lattice quantizer is one of the classical lattices, and there is good numerical evidence for this in dimensions 4 to 8. In contrast, in 9 dimensions, more than two decades ago, the same numerical studies found the smallest known value of G for a non-classical lattice. The structure and properties of this conjectured optimal lattice quantizer depend upon a real parameter (Formula presented.), whose value was only known approximately. Here, a full description of this one-parameter family of lattices and their Voronoi cells is given, and their (scalar and tensor) second moments are calculated analytically as a function of a. The value of a which minimizes G is an algebraic number, defined by the root of a 9th order polynomial, with (Formula presented.). For this value of a, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The structure of the Voronoi cell depends upon a, and undergoes phase transitions at (Formula presented.), 1, and 2, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. The methods can be used for arbitrary one-parameter families of laminated lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.

Keywords

    lattices, nine dimensions, optimal, quantizers

ASJC Scopus subject areas

Cite this

The Optimal Lattice Quantizer in Nine Dimensions. / Allen, Bruce; Agrell, Erik.
In: Annalen der Physik, Vol. 533, No. 12, 2100259, 14.12.2021.

Research output: Contribution to journalArticleResearchpeer review

Allen B, Agrell E. The Optimal Lattice Quantizer in Nine Dimensions. Annalen der Physik. 2021 Dec 14;533(12):2100259. Epub 2021 Oct 24. doi: 10.48550/arXiv.2104.10107, 10.1002/andp.202100259
Allen, Bruce ; Agrell, Erik. / The Optimal Lattice Quantizer in Nine Dimensions. In: Annalen der Physik. 2021 ; Vol. 533, No. 12.
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