Quasi-orthogonal cocycles and optimal sequences
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 ∈ Z2(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.