The optimal lattice quantizer in nine dimensions

The optimal lattice quantizer is the lattice which minimizes the (dimensionless) second moment $G$. In dimensions $1$ to $8$, it has been proven that the optimal lattice quantizer is one of the classical lattices, or there is good evidence for this. In contrast, more than two decades ago, convincing numerical studies showed that in dimension $9$, a non-classical lattice is optimal. The structure and properties of this lattice depend upon a real parameter $a>0$, whose value was only known approximately. Here, we give a full description of this one-parameter family of lattices and their Voronoi cells, and calculate their (scalar and tensor) second moments analytically as a function of $a$. The value of $a$ which minimizes $G$ is an algebraic number, defined by the root of a $9$th order polynomial, with $a \approx 0.573223794$. 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 $a^2 = 1/2$, $1$ and $2$, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. Our methods can be used for arbitrary one-parameter families of layered lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.