Cocyclic Orthogonal Designs and the Asymptotic Existence of Cocyclic Hadamard Matrices and Maximal Size Relative Difference Sets with Forbidden Subgroup of Size 2

Abstract

This 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 by Z2 are shown to act regularly on the associated group divisible design of the Sylvester Hadamard matrices. Cocyclic orthogonal designs are then introduced, and the construction and classification questions for cocyclic M-concordant systems of orthogonal designs are addressed. (M-concordance generalises the concepts of amicability and anti-amicability.) We give an algebraic procedure for constructing and classifying these designs when each indeterminate is constrained to appear just once in each row and column of the orthogonal designs. This paper also gives a general (but apparently not comprehensive) method for constructing systems with no zero entries. In particular, we obtain a cocyclic pair of amicable OD(16; 8, 8). Using this pair of designs, we prove there is a cocyclic Hadamard matrix of order 2ts for any odd integer s>1 and any t⩾⌊8log2s⌋. A consequence of our argument is the theorem, “Let S be any group of odd order s containing k prime order subgroups Si of S such that (1) for i=0, …, k−1, the sets Ui=Si+1Si+2…Sk are subgroups of S, (2) for i=0, …, k−1, Si∩Ui=〈1〉, and (3) U0=S. Let T be any group of order 2t+1 containing a central involution z such that (1) T/〈z〉 is elementary abelian, and (2) the largest elementary abelian direct factor of T has rank between 4log2s and t−2−4log2s. Then there is a normal relative difference set of size 2ts with forbidden subgroup 〈z〉 in the group T×S.” The condition on S holds for any group of odd square-free order or any direct product of odd order elementary abelian groups.