Parametric Lattices Are Better Quantizers in Dimensions 13 and 14

Abstract

New lattice quantizers with lower normalized second moments than previously reported are constructed in 13 and 14 dimensions and conjectured to be optimal. Our construction combines an initial numerical optimization with a subsequent analytical optimization of families of lattices, whose Voronoi regions are constructed exactly. The new lattices are constructed from glued products of previously known lattices, by scaling the component lattices and then optimizing the scale factors. A two-parameter family of lattices in 13 dimensions reveals an intricate landscape of phase changes as the parameters are varied.

Figures (7)

• Figure 1: Theta image of a numerically optimized $14$-dimensional lattice.
• Figure 2: NSM for the $14$-dimensional lattice \ref{['eq:gen14']} as function of $a$. The dot marks the location of the minimum and the dashed vertical lines indicate the boundaries of the phase \ref{['eq:phase14']}. The previously reported minimum NSM of $0.069\,52$lyu22 is about eight plot-heights above the dot.
• Figure 3: Theta image of the conjectured optimal $14$-dimensional lattice, rescaled to determinant 1 and superimposed on the theta image of the numerically optimized lattice in Fig. \ref{['f:theta14']}. The three shells to the left of the gray bar were used to determine the parametric lattice from the numerical one.
• Figure 4: Theta image of a numerically optimized $13$-dimensional lattice.
• Figure 5: Phases A and B of the lattice \ref{['eq:gen13']}. Solid lines show phase boundaries where $1$-face lengths vanish and the dots mark the points where lines of vanishing $1$-face lengths intersect. The color gradient indicates the value of $\Delta G({\boldsymbol{a}}) \triangleq G({\boldsymbol{a}}) - G({\boldsymbol{a}_{\text{opt}}})$, where $G = G_A$ in phase A, $G = G_B$ in phase B, and $G({\boldsymbol{a}_{\text{opt}}})$ is given in \ref{['eq:Gopt13']}. The functions $G_B$ and $G_A$ are extremely similar throughout the colored region (see \ref{['eq:NSM13B']} and \ref{['eq:DG']}). Dashed lines show contours of equal $G$. The dotted arrows indicate the search path described in Sec. \ref{['sub:opt13']}. The minimum of $G$ lies very close to ${\boldsymbol{a}}_B$ (see \ref{['eq:aA_vs_aB']}).