Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension and its exact normalized second moment is computed.

40 years ago, Conway and Sloane proposed using the highly symmetrical Coxeter–Todd latticeK12for quantization, and estimated its second moment. Since then, all published lists identifyK12as the best 12-dimensional lattice quantizer. Surprisingly,K12is not optimal: we construct two new 12-dimensional lattices with lower normalized second moments. The new lattices are obtained by gluing together products of two 6-dimensional lattices.