  A lattice quantizer approximates an arbitrary real-valued source vector with
a vector taken from a specific discrete lattice. The quantization error is the
difference between the source vector and the lattice vector. In a classic 1996
paper, Zamir and Feder show that the globally optimal lattice quantizer (which
minimizes the mean square error) has white quantization error: for a uniformly
distributed source, the covariance of the error is the identity matrix,
multiplied by a positive real factor. We generalize the theorem, showing that
the same property holds (i) for any lattice whose mean square error cannot be
decreased by a small perturbation of the generator matrix, and (ii) for an
optimal product of lattices that are themselves locally optimal in the sense of
(i). We derive an upper bound on the normalized second moment (NSM) of the
optimal lattice in any dimension, by proving that any lower- or
upper-triangular modification to the generator matrix of a product lattice
reduces the NSM. Using these tools and employing the best currently known
lattice quantizers to build product lattices, we construct improved lattice
quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions,
these are the first reported lattices with normalized second moments below the
best known upper bound.
