Barnes-Wall Lattices Research Articles

BASIC SPIN REPRESENTATIONS OF ALTERNATING GROUPS, GOW LATTICES, AND BARNES-WALL LATTICES

In a recent paper, R. Gow showed that in certain cases the basic spin representations of the group (of degree ) can be rational. In such cases, the -invariant lattices in the corresponding rational module have many interesting properties. In the present paper all possibilities are found for the groups . Also, a conjecture of Gow is proved: For , , there is among the -invariant lattices the even unimodular Barnes-Wall lattice . At the same time, the rationality of the basic spin representation of and the reducibility of as a -module are proved. Bibliography: 21 titles.

New decoding strategy for the 32-dimensional Barnes-Wall lattice

For QAM signalling, a new decoding strategy which reduces the average decoding complexity of the 32-dimensional Barnes–Wall lattice is presented. The reduction is by an order of magnitude at an SNR per bit of 10.5 dB. The strategy is based on partitioning the 32-D code space into disjoint subsets. The average decoding complexity is highly dependent on the number of these subsets.

The Coset lattices of E. S. Barnes and G. E. Wall

AbstractJohn Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups. (The classification theorem for the finite simple groups has since told us that no new finite simple groups can arise in this or any other way.) For example in 1973, M. Broué and M. Enguehard constructed, in every dimension 2n, an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type Dn. This family of integral lattices received attention and acclaim in the subsequent literature. What escaped the attention of this literature, however, was the fact that these lattices had been discovered years earlier. Indeed in 1959, E. S. Barnes and G. E. Wall gave a uniform construction for a large class of positive definite Z-lattices in dimensions 2n which include those of Broué and Enguehard as special cases. The present article introduces an abstracted and generalized version of the construction of Barnes and Wall. In addition, there are some new observations about Barnes-Wall lattices. In particular, it is shown how to associate to each such lattice a continuous, piecewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.

Lattice instantons in the large dimension limit

We give a lattice construction of the discretizations of the topologically nontrivial maps S2n−1→S n . For n=1, 2, 4, 8, these are the Hopf maps. The construction, based on Barnes-Wall lattices, Reed-Muller error-correcting codes, and Hadamard matrices, generalizes to n=2 i for i any integer. Manton's result for the cases n=2 and 4 (i.e., the monopole and instanton) are included. We argue that discrete harmonic analysis will be exact in the infinite dimension limit.

Coset codes. II. Binary lattices and related codes

For pt.I see ibid., vol.34, no.5, p.1123-51 (1988). The family of Barnes-Wall lattices (including D/sub 4/ and E/sub 8/) of lengths N=2/sup 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 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*2/sup 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. >