Guest: Tekgül Kalaycı, Sabancı University
Title: Bent partitions and related combinatorial objects
Date/Time: April 28, 2025, 14:40
Location: FENS G015
Abstract:Recently, the concept of bent partitions was introduced by Anbar and Meidl, 2022, which are partitions of elementary abelian groups having similar properties as spreads. A partition Ω of a 2m-dimensional vector space V(p) 2m over Fp, into an m-dimensional subspace U, and subsets A1, . . . , AK, is called a bent partition of depth K if every function from V(p) 2m to Fp with the following properties, is a bent function: Every c ∈ Fp has precisely K/p of the sets Aiin its preimage set f−1(c) = {x ∈ V(p) 2m : f(x) = c} and f is constant on U. From (pre)semifields whose duals satisfy a linearity property over a subfield of Fpm, we construct a large class of bent partitions of Fpm × Fpm, which we call generalized semifield spreads. We show that generalized semifield spreads do not arise from any spread and one obtains not only p-ary and vectorial bent functions, but also bent functions into finite abelian groups of certain orders from generalized semifield spreads. We further show that any union of the sets in generalized semifield spreads forms a partial difference set. Moreover, such a partition induces an edge decomposition of the complete graph with vertex set Fpm × Fpm into strongly regular graphs, where any union of the graphs remains strongly regular. This enables us to construct (pk + 1)-class amorphic association schemes on Fpm × Fpm from (pre)semifields, where k is a divisor of m with special properties. A vectorial bent function into an m-dimensional vector space over Fp is called a vectorial dual-bent function if the set of duals of its component functions forms an m-dimensional vector space of bent functions together with the zero function. We point out relations between bent partitions, vectorial dual-bent functions, and Latin square partial difference set packings (LP-packings), a concept introduced by Jedwab and Li, 2022. Finally, we present constructions of bent partitions that do not yield partial difference sets and strongly regular graphs, and do not form LP-packings.
Bio: Tekgül Kalaycı received her B. S. degree in Mathematics from Dokuz Eylül University, İzmir, in 2011, and her M.S. degree in Mathematics from the İzmir Institute of Technology in 2014. She completed her Ph.D. in January 2019 at Sabancı University, İstanbul, under the supervision of Prof. Alev Topuzoğlu. Following her Ph.D., she held postdoctoral positions at Sabancı University, working with Prof. Cem Güneri on coding theory and with Dr. Nurdagül Anbar Meidl on cryptographically significant functions and related combinatorial structures. She is currently a postdoctoral researcher at the University of Klagenfurt, under the supervision of Dr. Wilfried Meidl. During her postdoctoral research, she has had ongoing research visits to Prof. Alexander Pott at Otto-von-Guericke University in Magdeburg, Germany and Prof. Enes Pasalic at the University of Primorska in Koper, Slovenia. Her research interests include permutation polynomials, polynomial factorization, coding theory, and cryptographically significant functions—such as functions with high nonlinearity, almost perfect nonlinear functions, and bent and generalized bent functions—and their connections to combinatorial objects, including designs, difference sets, and their generalizations, such as partial difference sets and partial difference families, as well as graphs, packings, and association schemes.