The BCC lattice is the optimal 3D lattice quantizer for all $p\geq1$
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Senping Luo · arXiv: 2611.09523v1 · Summary page
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The best lattice for rounding space
Quantization replaces a continuous quantity with a nearby representative: an operation underlying digital measurement, compression and numerical computation. A new preprint proves that the body-centered cubic lattice is the uniquely optimal spatial lattice quantizer, up to rotations and reflections, for every finite distance-moment exponent p≥1. The comparison fixes the volume available per lattice point, so competing arrangements receive the same spatial budget.
The mathematical objective measures distances within a lattice's Voronoi cell, the region assigned to a particular lattice point. Establishing optimality across this whole family of distance penalties proves the Graf–Luschgy conjecture for spatial lattice quantizers. It gives a firm theoretical foundation for this arrangement, although a result about lattices is not a claim that every unrestricted compression scheme must use one.
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