**Quasi-orthogonal cocycles and optimal sequences**

### Dane Flannery,

**National University of Ireland, Galway **

Date: Friday 30th August

Time: 3:00pm–4:00pm (Talk & Q/A)

Venue: Building 8 Level 9 Room 66 (AGR) RMIT City campus

The seminar will be followed by snacks and drinks

All students, staff and visitors are welcome

**ABSTRACT:** *Orthogonal cocycles* arise in the study of symmetry of pairwise combinatorial designs. In the simplest and best known case, these are cocycles, ψ ∈ *Z*^{2}(*G*,<-1>), for a group, *G,* (of order divisible by *4*) whose display table [ψ(*g,h*)]_{g,h ∈ G} is a Hadamard matrix. A natural analogue is

*quasi-orthogonal cocycle*, defined over *G* of order congruent to *2* modulo *4*.

There is a connection to the maximal determinant problem (for example, if a cocyclic binary matrix of order *4t+2* attains the maximal determinant

bound then the cocycle is quasi-orthogonal). Quasi-orthogonal cocycles seem to be far more prevalent than the orthogonal kind.

We survey some recent results on quasi-orthogonal cocycles. These encompass new and known combinatorial objects: quasi-Hadamard groups, relative quasi-difference sets, and partially balanced incomplete block designs. In another direction, we note that generalized perfect binary arrays are known to be cocyclic; generalized optimal binary arrays are the relevant quasi-cocyclic analogue, and these lead to a new construction of negaperiodic Golay pairs. The next step is to widen the coefficient group from <-1>, obtaining (for example) quaternary sequences of odd length with optimal autocorrelation.

We also advertise a few prominent open problems that appear to be tractable. Apart from the obvious (existence), one of these concerns transposability of quasi-cocyclic matrices.

This is joint work with Josè Andrès Armario, University of Seville.

**BIOGRAPHY**: Dane Flannery is a Professor of Mathematics at National University of Ireland, Galway. His research interests range over algebra (particularly linear group theory), combinatorics and computing. He is a long-term collaborator of Emeritus Professor Kathy Horadam, and co-authored the 2011 monograph `Algebraic Design Theory’ with Dr Warwick de Launey.