A fast encoding method for lattice codes and quantizers

In an earlier paper the authors described a very fast method which, for the root lattices A₍, D₍, E₍, their duals and certain other lattices, finds the closest lattice point to an arbitrary point of the underlying space. If the lattices are used as codes for a Gaussian channel, the algorithm provides a fast decoding procedure, or if they are used as vector quantizers the algorithm performs the analog-to-digital conversion efficiently. The present paper offers a solution to the inverse problem for the same lattices (the encoding problem for channel codes or the digital-to-analog part of quantizing), namely, given an integer k, to find the kth code vector, and to the closely related problem of finding the index k of a given code vector.