arXiv:2605.05202v2 Announce Type: replace Abstract: We optimize the bit-to-symbol map of the $D_4$ and $E_8$ root lattices in reference to bit error rate, by implementing two searches on a performance metric, Hamming Density. The first search is a traditional gradient descent and the second is an efficient search using twisted theta functions. We hold other sources of lattice gain constant by fixing the lattice constellation, and consider basis changes that permute the integer labelings of the lattice points. After searching the possible basis matrices for $D_4$ and $E_8$, we found up to 0.09 dB of gain in $D_4$ bit error rate curves, and up to 0.36 dB of gain in $E_8$ compared to the standard bases commonly used in literature at a BER of $10^{-4}$, which is on the scale of gain from Gray coding uncoded QAMs.