Antisymmetry lattice

The same considerations imply that the number of three-dimensional lattices should increase, and this is found to be so. There are a total of 36 magnetic lattices, made up of 22 antisymmetry lattices, together with the 14 neutral Bravais lattices. 

Arrays with rotational antisymmetry have attractive properties for applications such as coded aperture imaging (e.g. Cook et al. 1984). Finger and Prince (1985) showed that all antisymmetric URAs could be derived from skew-Hadamard cyclic difference sets. Since all two dimensional lattices have 180 symmetry, all skew-Hadamard cyclic difference sets can generate both hexagonal and rectangular URA s with 180 antisymmetry. A subclass of these with order v (the number of cells in the basic pattern) equal to a prime p with p — 1 mod 12 were shown to generate a hexagonal array (HURA) with an additional 60 rotational antisymmetry. 

Modified uniformly redundant arrays (MURAs) having imaging properties similar to URAs were discussed by Gottesman and Fenimore (1989), and later Byard (1992) showed that a subset of these with v = p with the prime p = 5 mod 8 or with u = with the prime p = 3 mod 4 had 90 rotational antisymmetry on a rectangular lattice. We will consider antisymmetric redundant arrays (ARAs) to include both the skew-Hadamard arrays and the anti-symmetric MURAs. 

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