In this survey, we revisit the Rothaus paper and the chapter of Dillon's thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.

Four decades of research on bent functions

Linear sets in finite projective spaces

In this article I survey some recent results on planar functions, almost planar functions and modified planar functions from the perspective of difference sets.

Almost perfect and planar functions

Zusammenfassung.- Introduction.- Fundamentals.- General results and methods.- Applications.- Bibliopraphy.- Index.- List of tables.

Blocks of Finite Groups and Their Invariants

For each rank metric code $$\mathcal {C}\subseteq \mathbb {K}^{m\times n}$$C⊆Km×n, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $$\mathcal {C}$$C is $$\mathbb {K}$$K-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When $$\mathbb {K}$$K is a finite field $$\mathbb {F}_q$$Fq and $$\mathcal {C}$$C is a maximum rank distance code with minimum distance $$d<\min \{m,n\}$$d

On kernels and nuclei of rank metric codes

Preface and Acknowledgments An Overview Translation Plane Structure Theory Partial Spreads and Translation Nets Partial Spreads and Generalizations Quasifields Derivation Frequently Used Tools Sharply Transitive Sets SL(2, p) x SL(2, p)-Planes Classical Semifields Groups of Generalized Twisted Field Planes Nuclear Fusion in Semifields Cyclic Semifields T-Cyclic GL(2, q)-Spreads Cone Representation Theory Andre Net Replacements and Ostrom-Wilke Generalizations Foulser's ?-Planes Regulus Lifts, Intersections over Extension Fields Hyper-Reguli Arising from Andre Hyper-Reguli Translation Planes with Large Homology Groups Derived Generalized Andre Planes The Classes of Generalized Andre Planes C-System Nearfields Subregular Spreads Fano Configurations Fano Configurations in Generalized Andre Planes Planes with Many Elation Axes Klein Quadric Parallelisms Transitive Parallelisms Ovoids Known Ovoids Simple T-Extensions of Derivable Nets Baer Groups on Parabolic Spreads Algebraic Lifting Semifield Planes of Orders q4, q6 Known Classes of Semifields Methods of Oyama and the Planes of Suetake Coupled Planes Hyper-Reguli Subgeometry Partitions Groups on Multiple Hyper-Reguli Hyper-Reguli of Dimension 3 Elation-Baer Incompatibility Hering-Ostrom Elation Theorem Baer-Elation Theory Spreads Admitting Unimodular Sections-Foulser-Johnson Theorem Spreads of Order q2-Groups of Order q2 Transversal Extensions Indicator Sets Geometries and Partitions Maximal Partial Spreads Sperner Spaces Conical Flocks Ostrom and Flock Derivation Transitive Skeletons BLT-Set Examples Many Ostrom-Derivates Infinite Classes of Flocks Sporadic Flocks Hyperbolic Fibrations Spreads with Many Homologies Nests of Reguli Chains Multiple Nests A Few Remarks on Isomorphisms Flag-Transitive Geometries Quartic Groups in Translation Planes Double Transitivity Triangle Transitive Planes Hiramine-Johnson-Draayer Theory Bol Planes 2/3-Transitive Axial Groups Doubly Transitive Ovals and Unitals Rank 3 Affine Planes Transitive Extensions Higher-Dimensional Flocks j...j-Planes Orthogonal Spreads Symplectic Groups-The Basics Symplectic Flag-Transitive Spreads Symplectic Spreads When Is a Spread Not Symplectic? When Is a Spread Symplectic? The Translation Dual of a Semifield Unitals in Translation Planes Hyperbolic Unital Groups Transitive Parabolic Groups Doubly Transitive Hyperbolic Unital Groups Retraction Multiple Spread Retraction Transitive Baer Subgeometry Partitions Geometric and Algebraic Lifting Quasi-Subgeometry Partitions Hyper-Regulus Partitions Small-Order Translation Planes Dual Translation Planes and Their Derivates Affine Planes with Transitive Groups Cartesian Group Planes-Coulter-Matthews Planes Admitting PGL(3, q) Planes of Order = 25 Real Orthogonal Groups and Lattices Aspects of Symplectic and Orthogonal Geometry Fundamental Results on Groups Atlas of Planes and Processes Bibliography Theorems Models General Index

Handbook of Finite Translation Planes

Andre's theory of spreads spreads in PG(3,K) partial spreads and translation nets spreadsheets and partial spreadsets geometry of spreadsets co-ordinatization by spreadsets - general cases partial quasifields co-ordinatization by (partial) quasifields rational desarguesian nets quasigroups, loops and nuclei (pre)quasifields -algebraic axioms and autopisms the kernel of spreadsets and quasifields quadratics of two dimensional quasifields - Hall systems spreads in projective spaces kernel subplanes across Desarguesian nets derivation of finite spreads Foulser's covering theorem structure of Baer groups Frobenius complements, p-primitive collineations, and Klein-4 groups large planar groups finite generalized Andre systems and nearfields elation net theory Baer-elation theory semifields. (Part contents)

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Foundations of translation planes

It is shown that, if D is a nontrivial 2-(v,k,λ) symmetric design, with (k,λ)=1, admitting a flag-transitive automorphism group G of affine type, then v=pd, p an odd prime, and G is a point-primitive, block-primitive subgroup of AΓL1(pd). Moreover, O(G) acts flag-transitively, point-primitively on D, and D is isomorphic to the development of a difference set whose parameters and structure are also provided.

On Flag-Transitive Symmetric Designs of Affine Type

Nonsymmetric 2-(v,k,λ) designs, with (r,λ) = 1, admitting a solvable flag-transitive automorphism group of affine type

2-$$(v,k,1)$$(v,k,1) Designs with a point-primitive rank 3 automorphism group of affine type are investigated and several new examples are provided.

Mauro Biliotti

Papers

Handbook of Finite Translation Planes

Foundations of translation planes

On Flag-Transitive Symmetric Designs of Affine Type

Nonsymmetric 2-(v,k,λ) designs, with (r,λ) = 1, admitting a solvable flag-transitive automorphism group of affine type

2-$$(v,k,1)$$(v,k,1) Designs with a point-primitive rank 3 automorphism group of affine type