Higher-dimensional Hadamard matrices

The concept of a Hadamard matrix as a binary orthogonal matrix is extended to higher dimensions. An -dimensional Hadamard matrix is defined as one in which all parallel -dimensional layers, in any axis-normal orientation, are uncorrelated. This is equivalent to the requirements that and that where represents all permutations of . A "proper" -dimensional Hadamard matrix is defined as a special case of the above in which all two-dimensional layers, in all axis-normal orientations, are Hadamard matrices, as a consequence of which all intermediate-dimensional layers are also Hadamard matrices. Procedures are described for deriving three- and four-dimensional Hadamard matrices of varying propriety from two-dimensional Hadamard matrices. A formula is given for a fully proper -dimensional matrix of order two, which can be expanded by direct multiplication to yield proper Hadamard matrices. It is suggested that proper higher dimensional Hadamard matrices may find application in error-correcting cedes, where their hierarchy of orthogonalitias permit a variety of checking procedures. Other types of Hadamard matrices may be of use in security codes on the basis of their resemblance to random binary matrices.