Lattice labeling algorithms for vector quantization

Due to its regular structure, lattice vector quantization (VQ) often offers substantial reduction in complexity over conventional VQ. Essentially there are two issues involved in designing a lattice vector quantizer: the development of fast algorithms for lattice decoding, and the construction of efficient algorithms for lattice labeling. In labeling lattices, it is necessary to define a boundary within which the points are to be labeled. In this paper, the primary concern is on the development of lattice labeling algorithms with respect to pyramid boundaries. In particular, we have developed algorithms for labeling two large categories of lattices: the Construction-A and the Construction-B lattices including important lattices such as the Gosset lattice E₈ and the 16-dimensional Barnes-Wall lattice A₁₆. The algorithms are noted to achieve 100% efficiency in utilizing index bits for binary representations. Furthermore, it is determined that many important lattices (E₈, A₁₆, etc.) can be indexed to arbitrary norms and dimensions. The complexity of these algorithms with regard to both memory and computation are quite low and thus enable the development of practical lattice vector quantizers of large norms and high dimensions