Theory and Applications of Finite Fields.

Intro -- Preface -- Low dimensional models of the finite split Cayley hexagon -- 1. Introduction -- 2. The 3-dimensional Hermitian surface and its Baer substructures -- 3. The connection with the 6-dimensional parabolic quadric -- 4. Characterising the split Cayley hexagon in the 6-dimensional parabolic quadric -- 5. A connection with Phan theory -- Acknowledgements -- References -- Davenport's constant for groups with large exponent -- 1. Introduction and results -- 2. Systems of disjoint zero-sums -- 3. Proof of Theorem 1.1 -- 4. Proof of Theorem 1.4: The case ≤7 -- 5. Proof of Theorem 1.4: The case ≥7 -- References -- Permanent has less zeros than determinant over finite fields -- 1. Introduction -- 2. Matrices with zero permanent. -- Acknowledgments -- References -- On a series of modules for the symplectic group in characteristic 2 -- 1. Introduction -- 2. Proof of Theorem 1.2 -- Acknowledgments -- References -- Exact divisibility of exponential sums and some consequences -- 1. Introduction -- 2. Preliminaries -- 3. Exact Divisibility of Exponential Sums in One Variable over _{ } -- Acknowledgements -- References -- Additive character sums of polynomial quotients -- 1. Introduction -- 2. Small degree -- 3. Monomials -- Acknowledgment -- References -- 5-Designs related to binary extremal self-dual codes of length 24 -- 1. Introduction -- 2. The case =5 -- 3. Automorphism groups -- 4. Questions -- References -- Sequences of Dedekind sums in function fields -- 1. Introduction -- 2. Additive functions -- 3. Inhomogeneous Dedekind sums -- 4. Higher dimensional Dedekind sums -- 5. Proofs of Theorems 3.4, 4.6 -- 6. Limits of Dedekind sums -- References -- Niho Bent Functions and Subiaco Hyperovals -- 1. Introduction and Preliminaries -- 2. Subiaco Hyperovals -- 3. Bent Functions from Subiaco Hyperovals -- References.

A bound on the number of points of a curve in a projective space over a finite field -- 1. Introduction -- 2. Combinatorial approach -- 3. Number of points of a non-degenerate irreducible curve -- 4. Proof of Theorem 1.1 and Corollary 1.2 -- 5. Asymptotic behavior -- References -- Designs in Projective Hjelmslev Spaces -- 1. Introduction -- 2. Modules over Finite Chain Rings -- 3. Projective Hjelmslev Spaces -- 4. Relations between the -designs -- 5. Spreads -- 6. The Nonexistence of Some Spreads of Non-free Submodules -- 7. Open Problems -- Acknowledgements -- References -- On the nuclei of a finite semifield -- 1. Introduction -- 2. Isotopy relation and nuclei -- 3. The known families of commutative semifields -- 4. The isotopy issue -- References -- Small-bias sets from extended norm-trace codes -- 1. Introduction and preliminaries -- 2. Balanced codes and small-bias sets -- 3. Extended norm-trace codes and associated -biased sets -- References -- On the Waring Problem with multivariate Dickson polynomials -- 1. Introduction -- 2. Preparations -- 3. Existence of ₁( , , ) -- 4. Estimates for ₐ( , , ) -- 5. Final remarks -- References -- Polynomials modulo and the theory of Galois sets -- Introduction -- 1. Background -- 2. A Proof of Kronecker's Average value Theorem -- 3. Irreducibility of Polynomials Modulo -- 4. The Theory of Galois Sets -- References -- Additive decompositions induced by multiplicative characters over finite fields -- 1. Introduction -- 2. Preliminaries -- 3. Results -- 4. Connections with other problems -- References -- Graphs associated with the map \ + ⁻¹ in finite fields of characteristic two -- 1. Introduction -- 2. Preliminaries -- 3. The structure of the group of the rational points of an elliptic curve over a finite field -- 4. The structure of the graphs -- 5. Examples.

6. A note on the construction of the graphs -- References.

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