Topics in Finite Fields.

Cover -- Title page -- Contents -- Preface -- New recursive construction of normal polynomials over finite fields -- 1. Introduction -- 2. Preliminaries -- 3. Irreducibility of Polynomial Compositions -- 4. Some construction of N-polynomials -- References -- Collineation groups strongly irreducible on an oval in a projective plane of odd order -- 1. Introduction -- 2. Preliminary results -- 3. Proof of Theorem 1.1 -- 4. Proof of Theorem 1.2 -- References -- On the solvability of certain equations over finite fields -- 1. Introduction -- 2. Preliminary Lemmas -- 3. Main Results -- 4. Applications -- Acknowledgement -- References -- On automorphism groups of binary linear codes -- 1. Background and notations -- 2. Cyclic group of order ( an odd prime) -- 3. Cyclic group of order 2 ( an odd prime) -- 4. Dihedral group of order 2 ( an odd prime) -- 5. Interaction between fixed codes -- 6. The automorphism group of an extremal self-dual code of length 72 -- References -- Extended differential properties of cryptographic functions -- 1. Introduction -- 2. Block cipher basics -- 3. Differential cryptanalysis -- 4. New bounds on the maximum probability of two-round differentials -- 5. Sboxes with multiplication-invariant derivatives -- 6. Conclusions -- Acknowledgements -- References -- A divisibility criterion for exceptional APN functions -- 1. Introduction -- 2. The state of the art -- 3. New Results -- 4. Preliminary lemmata -- 5. Proof of theorem 3.1 -- 6. Some applications -- References -- Non weakly regular bent polynomials from vectorial quadratic functions -- 1. Introduction -- 2. Fourier coefficients for quadratic functions -- 3. Generating vectorial quadratic functions with discrete Fourier transform -- 4. Non weakly regular bent polynomials -- References -- Strongly regular graphs arising from Hermitian varieties.

1. Introduction -- 2. Preliminary results -- 3. Cliques of Γᵣ*(ℋ) -- 4. Automorphism groups -- Acknowledgement -- References -- Generalized rank weights: A duality statement -- 1. Introduction -- 2. General properties for the generalized rank weights -- 3. Duality and generalized rank weights : proof of Theorem 1.3 -- References -- An upper bound for the number of Galois points for a plane curve -- 1. Introduction -- 2. Preliminaries -- 3. Proof -- References -- A generalization of the nonlinear combination generator -- 1. Introduction -- 2. Preliminaries -- 3. Results on linear recurring sequences -- 4. Sequences of the form ₁ ₁+\dotsb+ _{ } _{ } -- 5. Multiplication strong feedback shift registers -- References -- Dedekind sums with a parameter in function fields -- 1. Introduction -- 2. Review of higher-dimensional Dedekind sums -- 3. Higher-dimensional Dedekind sums with a parameter -- 4. Proof of Theorem 3.3 -- 5. Limit theorem -- 6. Apostol-Dedekind sums -- References -- Numbers of points of hypersurfaces without lines over finite fields -- 1. Introduction -- 2. The first step -- 3. Surface over ₄ of degree 4 in ℙ³ -- References -- Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4 -- 1. Introduction -- 2. Preparations -- 3. The Computer Classification -- 4. Computer-Free Constructions -- 5. Conclusion -- Acknowledgements. -- References -- A survey of permutation binomials and trinomials over finite fields -- 1. Introduction -- 2. Permutation Binomials -- 3. Permutation Trinomials -- References -- Computing class groups of function fields using stark units -- 1. Introduction -- 2. Stickelberger elements and Stark units -- 3. Algorithms for computing the divisor class groups -- 4. Euler systems from Stark units -- 5. Characterizing ideal class group using Kolyvagin systems.

6. Computation of the ideal class group -- 7. Ackowledgements -- References -- Finding primitive elements in finite fields of small characteristic -- 1. Introduction -- 2. Finding primitive elements -- 3. Acknowledgements -- References -- The coset leader and list weight enumerator -- 1. Introduction -- 2. Codes, arrangements and geometric lattices -- 3. Coset leader and list weight enumerator -- 4. The derived code -- 5. Coset leader weights and the derived code -- 6. MacWilliams type property for duality -- 7. Directions for further work -- References -- Wieferich past and future -- 1. The early history -- 2. The Crandall-Dilcher-Pomerance model -- 3. Another point of view: the Wieferich Conjecture, first version -- 4. The general setting for a conjecture of Wieferich type -- 5. Interlude: The question of robustness -- 6. The framed Wieferich conjecture -- 7. The case of elliptic curves and products thereof -- 8. The case of abelian varieties -- 9. CM examples -- 10. Big monodromy examples -- 11. Other Jacobians with big monodromy -- 12. A variant lattice for of Jacobians -- 13. The case of semiabelian varieties -- References -- Field reduction and linear sets in finite geometry -- 1. Introduction -- 2. Field reduction for projective spaces -- 3. Field reduction for classical polar spaces -- 4. Linear sets in projective spaces -- 5. Blocking sets and field reduction -- 6. Semifields and linear sets -- References -- Bent functions from spreads -- 1. Introduction -- 2. Notation and preliminaries -- 3. Bent functions whose restrictions to each multiplicative cosets \GF ^{⋆} ( ∈ ) are constant -- 4. Bent functions whose restrictions to the -spread \GF ( ∈ ) are linear -- 5. Bent functions whose restrictions to the multiplicative cosets \GF ^{⋆} ( ∈ ) are affine -- 6. Conclusion -- Acknowledgement -- References.

On the characterization of a semi-multiplicative analogue of planar functions over finite fields -- 1. Introduction -- 2. Periodicity properties of Costas arrays -- 3. Costas polynomials and direct product difference sets -- 4. Costas polynomials in extension fields -- Acknowledgement -- References -- A solution of an equivalence problem for semisimple cyclic codes -- 1. Introduction -- 2. Preliminaries -- 3. Proof of main results -- 4. Algorithm for cyclic codes equivalence testing -- Acknowledgements -- References -- On cross joining de Bruijn sequences -- 1. Introduction -- 2. Some definitions and known theorems -- 3. The main theorem -- 4. Applications -- Acknowledgment -- References -- Ambiguity and deficiency of reversed Dickson permutations -- 1. Introduction -- 2. Background material -- 3. Divisibility of binomial coefficients -- 4. Ambiguity and deficiency of reversed Dickson polynomials -- Acknowledgement -- References -- From near-bent to bent: A special case -- 1. Introduction -- 2. Preliminaries -- 3. Results -- 4. Conclusion -- References -- Back Cover.

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