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 1001.05032J. 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 \(T\times S\).

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 \(T\times S\).

Reviewer: Spencer P.Hurd (Charleston)

##### MSC:

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

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

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\textit{W. de Launey} and \textit{M. J. Smith}, J. Comb. Theory, Ser. A 93, No. 1, 37--92 (2001; Zbl 1001.05032)

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