On the quantization goodness of polar lattices

In this work, we prove that polar lattices, when tailored for lossy compression, are quantization-good in the sense that their normalized second moments approach $\frac{1}{2\pi e}$ as the dimension of lattices increases. It has been predicted by Zamir et al. \cite{ZamirQZ96} that the Entropy Coded Dithered Quantization (ECDQ) system using quantization-good lattices can achieve the rate-distortion bound of i.i.d. Gaussian sources. In our previous work \cite{LingQZ}, we established that polar lattices are indeed capable of attaining the same objective. It is reasonable to conjecture that polar lattices also demonstrate quantization goodness in the context of lossy compression. This study confirms this hypothesis.