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 noise: 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 locally optimal lattice quantizer and (ii) for an optimal product lattice, if the component lattices are themselves locally optimal. 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 Zador upper bound.

翻译：色调量度器近似任意的、 真实价值源矢量, 其矢量取自特定的离散量 。 量化误差是源矢量与拉蒂矢量的差别。 在1996年经典论文中, Zamir 和 Feder 显示, 全球最佳的 lattice 量度器( 将平均平方差最小化 ) 有白度噪声: 对于统一分布源来说, 差错的共差是身份矩阵, 乘以一个正真实系数。 我们推广了该词, 显示同一属性( i) 对于任何本地最佳的 lattice 量量器和( ii) 最佳产值值的差值。 如果组件本身是本地最佳的, Zatic 值是本地最佳的。 在任何维度的正常第二时刻( NSM ), 我们得出一个上层的上限, 通过证明对产品拉蒂埃的发电机矩阵的任何低度或上方形修改都会减少 NSM 。 使用这些工具, 并使用目前已知的最佳的 lattace 质质质质质质质质质质质质量器来构建产品底层, 、 13 和25 平面的平面的平至25 度为平面的平至25 。