Abstract
AbstractThe 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 , 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 . 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 , 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.
Highlights
Introduction and SummaryThese vertices define the Voronoi cell, in the sense that their con-Lattices are regular arrays of points in Rn
While the optimal lattice quantizer of the form (1.2) has a2 < 1∕2, we have studied how the lattice behaves for larger values
More than three decades have passed since the publication of Conway and Sloane’s wonderful book on sphere packings, lattices and groups,[1] and more than two decades since the third edition appeared
Summary
Lattices are regular arrays of points in Rn. They are obtained as arbitrary linear combinations of n linearly independent basis vectors, with integer coefficients. In the course of this work we have explored the one-parameter family of lattices generated by B, obtaining exact formulae for the second moments, volumes, and other properties of all of the subfaces Our approach can be used to evaluate the scalar and tensor second moments U and Uμν, the volume V and the dimensionless scalar second moment G for one-parameter families of lattices in any number of dimensions n. For this purpose, it should be sufficient to evaluate U and V for at most 2n + 3 distinct rational values of the parameter
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