A weak difference set construction for higher dimensional designs.

*(English)*Zbl 0838.05019Authors’ abstract: We continue the analysis of de Launey’s modification of development of designs modulo a finite group \(H\) by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result. We extend the characterization of all AEFs from the cyclic group case to the case where \(H\) is an arbitrary finite abelian group. We prove that our \(n\)-dimensional designs have the form \((f(j_1j_2\dots j_n))\), \(j_i\in J\), where \(J\) is a subset of cardinality \(|H|\) of an extension group \(E\) of \(H\). 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 order \(v\) for which \(|E|= 2v\). In particular, we construct proper higher dimensional Hadamard matrices for all orders \(4t\leq 100\), and conference matrices of order \(q+ 1\) where \(q\) is an odd prime power. We conjecture that such Hadamard matrices exist for all orders \(v\equiv 0\bmod 4\).

Reviewer: J.R.Seberry (Wollongong)

##### MSC:

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |

05B99 | Designs and configurations |

##### Keywords:

group development; higher dimensional designs; weak difference set; orthogonal designs; Hadamard matrices; weighing matrices; conference matrices
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\textit{W. de Launey} and \textit{K. J. Horadam}, Des. Codes Cryptography 3, No. 1, 75--87 (1993; Zbl 0838.05019)

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##### References:

[1] | S.S. Agaian, On three-dimensional Hadamard matrix of Williamson type, (Russian-Armenian summary),Akad. Nauk Armyan. SSR Dokl., Vol. 72, (1981), pp. 131-134. |

[2] | S.S. Agaian, A new method for constructing Hadamard matrices and the solution of the Shlichta problem,Sixth Hungarian Coll. Comb., 6-11, (1981), pp. 2-3. |

[3] | G. Berman, Families of generalised weighing matrices,Canad. J. Math., Vol. 30, (1978), pp. 1016-1028. · Zbl 0406.05015 |

[4] | K.S. Brown,Cohomology of Groups, Graduate Texts in Math. 87, Springer, New York, (1982). |

[5] | W. de Launey, (0, G)-Designs and Applications, Ph.D. Thesis, University of Sydney, (1987). |

[6] | W. de Launey, On the construction ofn-dimensional designs from 2-dimensional designs,Australas. J. Combin., Vol. 1, (1990), pp. 67-81. · Zbl 0758.05034 |

[7] | P. Delsarte, J.M. Goethals, and J.J. Seidel, Orthogonal matrices with zero diagonal, II,Canad. J. Math., Vol. 23, (1971), pp. 816-832. · Zbl 0219.05010 |

[8] | A.V. Geramita and J. Seberry,Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Lecture Notes in Pure and Appl. Math. 45, Dekker, New York, (1979). · Zbl 0411.05023 |

[9] | J. Hammer and J. Seberry, Higher dimensional orthogonal designs and Hadamard matrices II,Congr. Numer., Vol 27, (1979), pp. 23-29. |

[10] | J. Hammer and J. Seberry, Higher dimensional orthogonal designs and Hadamard matrices,Congr. Numer., Vol. 31, (1981), pp. 95-108. · Zbl 0515.05019 |

[11] | J. Hammer and J. Seberry, Higher dimensional orthogonal designs and applications,IEEE Trans Inform. Theory, Vol. IT-27, (1981), pp. 772-779. · Zbl 0475.05021 |

[12] | A. Hedayat and W.D. Wallis, Hadamard matrices and their applications,Ann. Statist., Vol. 6, (1978), pp. 1184-1238. · Zbl 0401.62061 |

[13] | K.J. Horadam and W. de Launey, Cocyclic development of designs, (preprint), (1992). · Zbl 0785.05019 |

[14] | J. Seberry, Higher dimensional orthogonal designs and Hadamard matrices, inCombinatorial Mathematics VII, Lecture Notes in Math. 829, Springer, New York, (1980), pp. 220-223. |

[15] | P.J. Shlichta, Three and four-dimensional Hadamard matrices,Bull. Amer. Phys. Soc., ser. 11, Vol. 16, (1971), pp. 825-826. |

[16] | P.J. Shlichta, Higher dimensional Hadamard matrices,IEEE Trans. Inform. Theory, Vol. IT-25, (1979), pp. 566-572. · Zbl 0436.05009 |

[17] | Yang, Yi Xian, The proofs of some conjectures on higher dimensional Hadamard matrices,Kexue Tongbao (English translation), Vol. 31 (24), (1986), pp. 1662-1667. |

[18] | Yang, Yi Xian, On the classification of 4-dimensional 2 order Hadamard matrices, (preprint in English), (1986). |

[19] | Yang, Yi Xian, Onn-dimensional 2 order Hadamard matrices, (preprint in English), (1986). |

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