Coset codes. II: Binary lattices and related codes. (English) Zbl 0665.94019

Forney, G. David jun.

IEEE Trans. Inf. Theory 34, No. 5, Part II, 1152-1187 (1989).

The family of Barnes-Wall lattices (including \(D_ 4\) and \(E_ 8)\) of lengths \(N=2^ n\) and their principal sublattices, which are useful in constructing coset codes, are generated by iteration of a simple construction called the “squaring construction.” 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 properties of iterated squaring constructions, which also exhibit the interrelationships between codes and lattices of different lengths. An extension called the “cubing construction” generates good codes and lattices of lengths \(N=3\cdot 2^ n\), including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related construction 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.

MSC:

94B15

Cyclic codes

Keywords:

Barnes-Wall lattices; squaring construction; Reed-Muller codes; distances; dimensions; partitions; generator matrices; duality properties; Golay code; Leech lattice; Nordstrom-Robinson code; maximum likelihood decoding algorithms