The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.
There are three parts.The first part (on orthogonal designs and systems of orthogonal designs) outlines some orthogonal design theory. It is recalled that the theory of Clifford Algebras may be used to relate the possible orders of amicable pairs of orthogonal designs to the numbers of variables involved.
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