Abstract
We continue the analysis of de Launey's modification of development of designs modulo a finite groupH by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result.
We extend the characterization of allAEFs from the cyclic group case to the case whereH is an arbitrary finite abelian group.
We prove that ourn-dimensional designs have the form (f(j 1 j 2 ...j n )) (j i ∈J), whereJ is a subset of cardinality |H| of an extension groupE ofH. We say these designs have a weak difference set construction.
We show that two well-known constructions for orthogonal designs fit this development scheme and hence exhibit families of such Hadamard matrices, weighing matrices and orthogonal designs of orderv for which |E|=2v. In particular, we construct proper higher dimensional Hadamard matrices for all orders 4t≤100, and conference matrices of orderq+1 whereq is an odd prime power. We conjecture that such Hadamard matrices exist for all ordersv≡0 mod 4.
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Communicated by D. Jungnickel
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De Launey, W., Horadam, K.J. A weak difference set construction for higher dimensional designs. Des Codes Crypt 3, 75–87 (1993). https://doi.org/10.1007/BF01389357
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DOI: https://doi.org/10.1007/BF01389357