# zbMATH — the first resource for mathematics

Cocyclic orthogonal designs and the asymptotic existence of cocyclic Hadamard matrices and maximal size relative difference sets with forbidden subgroup of size 2. (English) Zbl 1001.05032
J. Hadamard conjectured that there is a Hadamard matrix of order $$4t$$ for every positive integer $$t$$ and J. R. Seberry [J. Comb. Theory, Ser. A 21, 188-195 (1976; Zbl 0344.05009)] proved the asymptotic result: for every positive odd integer $$t$$ and any integer $$a> 2\log_2t$$, there is a Hadamard matrix of order $$2^at$$. More recently W. de Launey and K. J. Horadam conjectured that even more strongly cocyclic Hadamard matrices exist for all positive integral $$t$$ [Des. Codes Cryptography 3, 75-87 (1993; Zbl 0838.05019)].
The present authors extend the ongoing application of group theory to design theory and also continue the program of investigating these fundamental conjectures. This far-reaching paper contains a discussion of cocyclic Hadamard matrices, their associated relative difference sets, and regular group actions. Nearly all central extensions of the elementary abelian 2-groups are shown to act regularly on the associated group divisible design of the Sylvester Hadamard matrices.
There are too many new and interesting results to mention more than the following theorems:
(i) Let $$q$$ be any odd prime power, and suppose there are positive integers $$a$$ and $$b$$ such that $$2^{k+1}= a(q- 3)+ b(q+1)$$. Let $$K$$ be the group $$A_{k+1}$$ if $$q\equiv 1\pmod 4$$, and the group $$C_{k+1}$$ otherwise. Let $$z$$ be the central involution in $$K$$. Then there is a cocyclic Hadamard matrix with extension group $$E_q\times Z^{2k}_2\times K$$ and index group $$E_q\times Z^{2k}_2\times K/\langle z\rangle$$.
(ii) For any odd positive integer $$s$$, and any integer $$t\geq \lfloor 8\log_2s\rfloor$$, there exists a cocylic Hadamard matrix with index group $$E_s\times Z^t_2$$.
(iii) Let $$S$$ be any group of odd order $$s$$ which may be factored into prime order subgroups. Suppose that $$T$$ is any central extension of $$Z_2$$ by $$Z^t_2$$, and that $$T$$ contains a central elementary abelian 2-group with at least $$s^4$$ elements and an extra special 2-group with at leat $$8s^4$$ elements. If $$z$$ is the central involution in the extra special 2-group, then there is a normal relative difference set of size $$2^ts$$ with forbidden subgroup $$\langle z\rangle$$ in the group $$T\times S$$.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
Full Text:
##### References:
 [1] Dillon, J.F., Variations on a scheme of mcfarland for noncyclic difference sets, J. combin. theory A, 40, 9-20, (1985) · Zbl 0583.05016 [2] Flannery, D.L., Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. algebra, 192, 749-779, (1997) · Zbl 0889.05032 [3] K. J. Horadam, D. L. Flannery, and, W. de Launey, Cocyclic Hadamard matrices and difference sets, Discr. Appl. Math, to appear. · Zbl 0956.05026 [4] Geramita, A.V.; Seberry, J., Orthogonal designs forms and Hadamard matrices, (1979), Dekker New York · Zbl 0411.05023 [5] Hadamard, J., Resolution d’une question relative aux determinants, Bull. sci. math., 17, 240-246, (1893) · JFM 25.0221.02 [6] Hall, M., The theory of groups, (1976), Chelsea New York [7] Horadam, K.J.; de Launey, W., Cocyclic development of designs, J. algebraic combin., 2, 267-290, (1993) · Zbl 0785.05019 [8] Huppert, B., Endliche gruppen I, (1967), Springer-Verlag Berlin/Heidelberg · Zbl 0217.07201 [9] Ito, N., On Hadamard groups, J. algebra, 168, 981-987, (1994) · Zbl 0906.05012 [10] Ito, N., On Hadamard groups, II, J. algebra, 169, 936-942, (1994) · Zbl 0808.05016 [11] de Launey, W.; Horadam, K.J., A weak difference set construction for higher dimensional designs, Des. codes cryptogr., 3, 75-87, (1993) · Zbl 0838.05019 [12] Robinson, D.J., Applications of cohomology to the theory of groups, (), 46-80 [13] Seberry, J.; Yamada, M., Hadamard matrices, sequences, and block designs, (), 431-560 · Zbl 0776.05028 [14] Sims, C.C., Computation with finitely presented groups, (1994), Cambridge Univ. Press Cambridge/New York/Melbourne · Zbl 0828.20001 [15] Sylvester, J.J., Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers, Phil. mag., 34, 461-475, (1867) [16] Wallis, J.S., On the existence of Hadamard matrices, J. combin. theory ser. A, 21, 188-195, (1976) · Zbl 0344.05009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.