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.

翻译：格点量化器将任意实值源向量近似为取自特定离散格点的向量。量化误差是源向量与格点向量之间的差值。在1996年的一篇经典论文中，Zamir与Feder证明全局最优格点量化器（使均方误差最小化）具有白色量化误差：对于均匀分布的源，误差协方差矩阵为单位矩阵乘以一个正实数因子。我们推广该定理，证明相同性质适用于：(i) 任何其均方误差无法通过生成矩阵的微小扰动而降低的格点；(ii) 在(i)意义下自身局部最优的格点之最优乘积。通过证明对乘积格点的生成矩阵进行任何下三角或上三角修正均会降低其归一化二阶矩（NSM），我们推导出任意维度最优格点的归一化二阶矩上界。利用这些工具并采用当前已知最佳格点量化器构建乘积格点，我们在13至15维、17至23维及25至48维中构造出改进的格点量化器。在某些维度上，这些是首次报道的归一化二阶矩低于已知最佳上界的格点结构。