Introduction to Data Compression, 5th Edition

Appendix C

The Root Lattices

Define ${\mathbf{e}}_i^L$ to be a vector in L dimensions whose ith component is 1 and all other components are 0. Some of the root systems that are used in lattice vector quantization are given as follows:

D L	$\pm {\mathbf{e}}_i^L\pm {\mathbf{e}}_j^L$,	i ≠ j, i,j = 1,2,…,L
A L	$\pm ({\mathbf{e}}_i^{L+1}-{\mathbf{e}}_j^{L+1})$,	i ≠ j, i,j = 1,2,…,L
E L	$\pm {\mathbf{e}}_i^L\pm {\mathbf{e}}_j^L$,	i ≠ j, i,j = 1,2,…, ...