The Optimal Lattice Quantizer in Nine Dimensions

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 , 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.