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Higher Dimensional Orthogonal Designs and Applications

Journal Article


Abstract


  • The concept of orthogonal design is extended to higher dimensions. A proper g dimensional design [Formula Omeeted] is defined as one in which all parallel [Formula Omeeted]-dimensional layers, in any orientation parallel to a hyper plane, are uncorrelated. This is equivalent to the requirement that [Formula Omeeted] where [Formula Omeeted] are integers giving the occurrences of [Formula Omeeted] in each row and column (this is called the type [Formula Omeeted] and [Formula Omeeted] represents all permutations of [Formula Omeeted]. This extends an idea of Paul J. Shlichta, whose higher dimensional Hadamard matrices are special cases with [Formula Omeeted] Another special case is higher dimensional weighing matrices of type (k)g, which have [Formula Omeeted] and [Formula Omeeted] Shlichta found proper g-dimensional Hadamard matrices of size (2t)g. Proper orthogonal designs of type [Formula Omeeted] and [Formula Omeeted] are used to obtain higher dimensional orthogonal designs, Hadamard matrices, and weighing matrices. A possible approach to using higher dimensional weighing matrices and Hadamard matrices in codes is discussed, as well as their connection with higher dimensional orthogonal functions (Walsh, Haar, etc.). © 1981 IEEE

Publication Date


  • 1981

Citation


  • Hammer, J., & Seberry, J. R. (1981). Higher Dimensional Orthogonal Designs and Applications. IEEE Transactions on Information Theory, 27(6), 772-779. doi:10.1109/TIT.1981.1056426

Scopus Eid


  • 2-s2.0-0019633486

Web Of Science Accession Number


Start Page


  • 772

End Page


  • 779

Volume


  • 27

Issue


  • 6

Abstract


  • The concept of orthogonal design is extended to higher dimensions. A proper g dimensional design [Formula Omeeted] is defined as one in which all parallel [Formula Omeeted]-dimensional layers, in any orientation parallel to a hyper plane, are uncorrelated. This is equivalent to the requirement that [Formula Omeeted] where [Formula Omeeted] are integers giving the occurrences of [Formula Omeeted] in each row and column (this is called the type [Formula Omeeted] and [Formula Omeeted] represents all permutations of [Formula Omeeted]. This extends an idea of Paul J. Shlichta, whose higher dimensional Hadamard matrices are special cases with [Formula Omeeted] Another special case is higher dimensional weighing matrices of type (k)g, which have [Formula Omeeted] and [Formula Omeeted] Shlichta found proper g-dimensional Hadamard matrices of size (2t)g. Proper orthogonal designs of type [Formula Omeeted] and [Formula Omeeted] are used to obtain higher dimensional orthogonal designs, Hadamard matrices, and weighing matrices. A possible approach to using higher dimensional weighing matrices and Hadamard matrices in codes is discussed, as well as their connection with higher dimensional orthogonal functions (Walsh, Haar, etc.). © 1981 IEEE

Publication Date


  • 1981

Citation


  • Hammer, J., & Seberry, J. R. (1981). Higher Dimensional Orthogonal Designs and Applications. IEEE Transactions on Information Theory, 27(6), 772-779. doi:10.1109/TIT.1981.1056426

Scopus Eid


  • 2-s2.0-0019633486

Web Of Science Accession Number


Start Page


  • 772

End Page


  • 779

Volume


  • 27

Issue


  • 6