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Chapter 10:
Higher-Dimensional Hadamard Matrices

High-dimensional Hadamard matrices can be found in nature; e.g., a typical model of a rock salt crystal is a 3D Hadamard matrix of order 4 (see Fig. 10.1).

Higher-dimensional Hadamard matrices were introduced several decades ago. Shlichta was the first to construct examples of n-dimensional Hadamard matrices. He proposed the procedures for generating the simplest 3D, 4D, and 5D Hadamard matrices. In particular, he put special emphasis on construction of the "proper" matrices, which have a dimensional hierarchy of orthogonalities. This property is a key for many applications such as error-correction codes and security systems. Shlichta also suggests a number of unsolved problems and unproven conjectures, as follows:

• The algebraic approach to the derivation of 2D Hadamard matrices (see Chapters 1 and 4) suggests that a similar procedure may be feasible for 3D or higher matrices.

• Just as families of 2D Hadamard matrices (such as skew and Williamson matrices) have been defined, it may be possible to identify families of higher-dimensional matrices, especially families that extend over a range of dimensions as well as orders.

• An algorithm may be developed for deriving a completely proper n3 (or nm) Hadamard matrix from one that is n2.

• Two-dimensional Hadamard matrices exist only in orders of 1, 2, or 4t. No such restriction has yet been established for higher dimensions. There may be absolutely improper n3 or nm Hadamard matrices of order n = 2s ≠ 4t.

• Shlichta's work prompted a study of higher-dimensional Hadamard matrix designs. Several articles and books on higher-dimensional Hadamard matrices have been published. In our earlier work, we have presented the higher-dimensional Williamson-Hadamard, generalized (including Butson) Hadamard matrix construction methods, and also have introduced (λ, μ)-dimensional Hadamard matrices.

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