Coset Codes-Part 11: Binary Lattices and Related Codes
Jr . G. David Forney · 1988
The family of Barnes-Wall lattices (including D4 and E,) of lengths N = 2 and their principal sublattices, which are useful in con- structing coset codes, are generated by iteration of a simple called the The closely related Reed-Muller codes are generated by the same construction. The principal properties of these codes and lattices, including distances, dimensions, partitions, generator matrices, and duality properties, are consequences of the general proper- ties of iterated squaring constructions, which also exhibit the interrelation- ships between codes and lattices of different lengths. An extension called the cubing construction generates good codes and lattices of lengths N = 3.2, including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related generates the Nordstrom-Robinson code and an analogous 16-dimensional nonlattice packing. These constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms. General algebraic methods for determining minimal trellis diagrams of codes, lattices, and partitions are given in an Appendix.