مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

The concept of a Hadamard matrix as a binary orthogonal matrix is extended to higher dimensions. An $n$ -dimensional Hadamard matrix $[h_{ijk \\cdots n}]$ is defined as one in which all parallel $(n - 1)$ -dimensional layers, in any axis-normal orientation, are uncorrelated. This is equivalent to the requirements that $h_{ijk \\cdots n} = \\pm1$ and that $\\sum _{p} \\sum _{q} \\sum _{r} \\cdots \\sum _{y} h_{pqr \\cdots yb}= m^{(n-1)} \\delta _{ab}$ where $(pqr \\cdots yz)$ represents all permutations of $(ijk \\cdots n)$ . A "proper" $n$ -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 $n$ -dimensional matrix of order two, which can be expanded by direct multiplication to yield proper $(2^{t})^{n}$ 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.