Optimizing Bit-Labeling of Voronoi Constellations

We define a novel search method and performance metric as a technique for optimizing the bit-to-symbol map of the $D_4$ and $E_8$ root lattices in reference to bit error rate. We hold other sources of lattice gain constant by fixing the lattice constellation, and consider basis matrices that permute the integer labelings of the lattice points. After searching the possible basis matrices for $D_4$ and $E_8$, we found 0.1 dB of gain in $D_4$ bit error rate curves, and 0.5 dB of gain in $E_8$ compared to the standard bases commonly used in literature at a BER of $10^{-4}$.