Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

Abstract

Using cohomology we show that in studying the existence of an abelian non-splitting (4t, 2, 4t, 2t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets.

Generalized binary arrays from quasi-orthogonal cocycles

“…For each ψ ∈ Z 2 (G, Z 2 ) we have a canonical central extension E ψ with element set {(±1, g) | g ∈ G} and multiplication defined by (u, g)(v, h) = (uvψ(g, h), gh). 3 Generalized binary arrays with optimal autocorrelation Jedwab [14] showed that a GPBA is equivalent to an abelian relative difference set, and Hughes [11] identified its underlying orthogonal cocycle. In this section we carry over these ideas into the setting of quasi-orthogonal cocycles.…”

“…Jedwab [14] introduced generalized perfect binary arrays (GPBAs) to aid in the construction of PBAs. Hughes [11] subsequently demonstrated the cocyclic nature of GPBAs.…”

Equivalence classes of central semiregular relative difference sets

“…Hughes [13] shows that any abelian group E of even order containing a subgroup N Z 2 can be realised as a Jedwab group J, by an isomorphism preserving the speci®ed order two subgroups. Theorem 5.3 applies, and #Z 2 Y EY EaN 1.…”

“…Over the past decade, applying group cohomology to combinatorial design theory by exploiting the correspondence (e.g. [1,6,8,10,11,13,18]) has therefore largely consisted of theoretical and computational searching for orthogonal cocycles, using algorithms for computing the full group of cocycles. This has given rise to the perception that orthogonality is an essentially combinatorial property, with no natural cohomological interpretation.…”

“…Currently the only known semiregular RDS in groups E which are not p-groups, have jCj w 2 a or w 3 (refer [5, p.70]). When w 2, that is C Z 2 , many orthogonal cocycles, and hence many relative 4tY 2Y 4tY 2t-difference sets, are known [1,6,8,13,14], through their identi®cation with cocyclic Hadamard matrices. Classi®cation of abelian relative 4tY 2Y 4tY 2t-difference sets, which correspond to generalised perfect binary arrays, is suggested as a starting point.…”