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 numerical 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 single positive real parameter $a$, whose value was only known approximately. Here, for $a^2 < 1/2$, we give an exact analytic description of this one-parameter family of lattices and their Voronoi cells, and calculate their second moment, which is a $19$th order polynomial in $a$. This allows us to determine the exact value of $a$ which minimizes $G$. It is an algebraic number, defined by the root of a $9$th order polynomial, with $a \approx 0.573223794$. We also show that 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 same method can be used for arbitrary one-parameter families of laminated lattices, so may provide a useful tool to identify optimal quantizers in other dimensions as well.