CONTENTS -- PREFACE -- References -- CONSTRUCTIONS AND CHARACTERIZATIONS OF CLASSICAL SETS IN PG(n q) -- Abstract -- 1. Introduction -- 2. Classical Sets with Few Intersection Numbers in PG(2 q) -- 2.1. Conics, Ovals and Hyperovals -- 2.1.1. Known Hyperovals -- Remarks -- 2.1.2. Characterization Theorems of Conics and Related Sets -- 2.2. Maximal Arcs -- 2.2.1. Introduction -- 2.2.2. The Known Constructions of Maximal Arcs -- The Construction by R. Mathon -- The Construction by R. Denniston -- The Constructions by J. A. Thas -- Remark -- 2.2.3. Some Characterization Theorems for Maximal Arcs -- 2.2.4. Maximal Arcs in Small Desarguesian Planes -- 2.3. Hermitian Curves and Unitals -- 2.3.1. De nitions and Constructions -- Remarks -- 2.3.2. Characterization Theorems -- 2.4. Characterizing Subplanes of PG(2 q) -- Remark -- 3. Classical Sets with Few Intersection Numbers in PG(n q), n ≥ 3 -- 3.1. Quadrics and Quasi-quadrics -- 3.1.1. De nitions -- 3.1.2. Characterization Theorems -- − Remarks -- Remark -- 3.1.3. Ovoids and Generalizations -- 3.2. Hermitian Varieties -- 3.3. Subgeometries -- Open Problems -- References -- SUBSTRUCTURES OF FINITE CLASSICAL POLAR SPACES -- Abstract -- 1. Finite Classical Polar Spaces -- 2. Isomorphisms of Finite Classical Polar Spaces -- 3. Ovoids, Spreads, m-systems and m-ovoids -- 3.1. Ovoids -- 3.2. Spreads -- 3.3. m-Systems -- 3.4. m-Ovoids -- 4. Partial Ovoids and Partial Spreads -- 4.1. Partial Ovoids -- 4.2. Partial Spreads -- 5. Covers and Blocking Sets -- 5.1. Covers -- 5.2. Blocking Sets -- References -- BLOCKING SETS IN PROJECTIVE SPACES -- Abstract -- 1. Introduction and De nitions -- 2. History and Basic Bounds -- 3. Natural Constructions -- 3.1. Subgeometry -- 3.2. Cone and Projection -- 3.3. Directions and the Generalized R ́edei Construction -- 4. Linear Blocking Sets -- 5. More Constructions.

5.1. Planar Constructions -- 5.2. Sporadic Constructions in Higher Dimensions -- 5.3. More Constructions in Higher Dimensions -- 5.4. The Mazzocca, Polverino, Storme Constructions -- 5.5. Some Interesting Examples Obtained by the MPS Construction -- 6. Af ne Blocking Sets -- Acknowledgm ents -- References -- LARGE CAPS IN PROJECTIVE GALOIS SPACES -- 1. What Is a Cap? -- 2. Classical Examples -- 3. Exceptional Caps -- The Ternary Case -- When q 3 -- 4. The Link to Linear Codes -- 5. General Bounds -- 6. Recursive Constructions -- 7. Families of Caps in Fixed Dimension -- The Case of Projective Dimension d = 4 -- Projective Dimension d ≤ 5 over F5 -- Higher Dimensions -- 8. Concrete Bounds -- 9. The Atoms of Cap Theory -- The Complete 14-cap in PG(3 4) -- A 66-cap in PG(4,5) -- A 132-cap in PG 4 7 -- A 208-cap in PG(4 8) -- A 195-cap in PG(5 5) -- A 434-cap in PG(5 7) -- 10. An Asymptotic Problem -- 11. Additive Codes and Quantum Caps -- 12. A Problem in Additive Number Theory -- A Global Approach -- Acknowledgm ents -- References -- THE POLYNOMIAL METHOD IN GALOIS GEOMETRIES -- Abstract -- 1. Introduction -- 2. Combinatorial Nullstellensatz -- 3. Nullstellens ̈atze for Lower Dimensional Subspaces? -- 4. Lacunary Polynomials -- 5. Vector Spaces of Polynomials and Functions over Fq -- 6. Field Extensions as Vector Spaces -- 7. Algebraic Curves over Finite Fields -- 8. Resultant of Polynomials in Two Variables -- 9. Open Problems -- 10. Final Comments -- Acknowledgments -- References -- FINITE SEMIFIELDS -- 1. Introduction and Preliminaries -- 1.1. De nition and First Properties -- 1.2. Projective Planes and Isotopism -- 1.3. Spreads and Linear Sets -- 1.4. Dual and Transpose of a Semi eld, the Knuth Orbit -- 2. Semi elds: A Geometric Approach -- 2.1. Linear Sets and the Segre Variety -- 2.2. BEL-construction -- 3. Rank Two Semi elds.

4. Symplectic Semi elds and Commutative Semi elds -- 5. Rank Two Commutative Semi elds -- 5.1. Translation Generalized Quadrangles and Eggs -- 5.2. Semi eld Flocks and Translation Ovoids -- 6. Known Examples and Classi cation Results -- 6.1. Classi cation Results for Any q -- 6.2. Classi cation Results for Small Values of q -- 7. Open Problems -- References -- CODES OVER RINGS AND RING GEOMETRIES -- Abstract -- 1. Projective and Af ne Hjelmslev Spaces -- 2. Coordinate Hjelmslev Geometries -- 3. Multisets of Points in Projective Hjelmslev Geometries and Linear Codes over Finite Chain Rings -- 3.1. Multisets of Points in PHG(Rk R) -- 3.2. Linear Codes over Finite Chain Rings -- 3.3. Equivalence of Multisets of Points and Linear Codes -- 3.4. Some Classes of Codes De ned Geometrically -- 4. Arcs in Projective Hjelmslev Planes -- 4.1. The Maximal Arc Problem -- 4.2. A General Upper Bound on the Size of an Arc -- 4.3. Constructions for Arcs -- 4.4. (k,2)-Arcs -- 4.5. Dual Constructions -- 4.6. Constructions Using Automorphisms -- 4.7. Tables for Arcs in Geometries over Small Chain Rings -- 5. Blocking Sets in Projective Hjelmslev Planes -- 5.1. General Results -- 5.2. Rédei Type Blocking Sets -- Acknowledgments -- References -- GALOIS GEOMETRIES AND CODING THEORY -- Abstract -- 1.0 Linear Codes over Finite Fields -- 1.01. General De nitions -- 1.12. Automorphisms of Linear Codes -- 1.13. The Spectrum of a Linear Code -- 1.14. Generalized HammingWeights -- 2.0 Arcs in Galois Geometries -- 2.01. Multiarcs and Minihypers -- 2.02. Equivalence of Multisets -- 2.03. Arcs and Codes -- 2.14. Weight Hierarchy and Generalized Spectra for Arcs -- 2.25. Constructions for Arcs -- Sum of Multisets -- Restriction to a Subspace -- Projections of Arcs -- The Dual Construction for Arcs -- 3.0 Arcs and Linear MDS Codes -- 3.01. Introduction to Arcs and Linear MDS Codes.

3.72. The Largest Arcs in Galois Geometries -- 3.73. Arcs in PG(2 q) -- 3.124. Results in Higher Dimensions -- 3.175. Open Problems -- 4.0 Minihypers and the Griesmer Bound -- 4.01. A Geometrical Proof of the Griesmer Bound -- 4.12. Minihypers and the Belov-Logachev-Sandimirov Construction -- 5.0 Saturating Sets in Galois Geometries and Covering Radius -- 5.61. Open Problems -- 6.0 Extension Results -- 6.01. The Extension Result of Hill and Lizak -- 6.72. Diversity and Extendability -- 6.83. Extension Results Depending on Divisibility and Quasi-divisibility -- 7.0 Codes Arising from Incidence Matrices of Galois Geome- tries -- 7.01. Linear Codes De ned by Incidence Matrices of Galois Geometries -- 7.02. SmallWeight Codewords -- 8.0 A Geometrical Result Obtained via Linear Codes -- Acknowledgm ents -- References -- APPLICATIONS OF GALOIS GEOMETRY TO CRYPTOLOGY -- Abstract -- 1. Introduction -- 1.1. Cryptography -- 1.2. Galois Geometry in Cryptography -- 2. Secret Sharing Schemes -- 2.1. Model for Secret Sharing -- 2.2. Linear Secret Sharing Schemes -- 2.3. Ideal Secret Sharing Schemes -- 2.4. Ef cient Linear Secret Sharing Schemes -- 2.5. Speci c Families of Access Structures -- 2.6. Secret Sharing Schemes with Extended Capabilities -- 2.6.1. Multiplicative Linear Secret Sharing Schemes -- 2.6.2. Multisecret Sharing Schemes -- 3. Authentication Codes -- 3.1. A-codes -- 3.2. A2-codes -- 3.3. Research Approaches -- 3.4. Geometric Constructions -- 4. Key Predistribution Schemes -- 4.1. Requirements -- 4.2. KPSs Based on Geometry -- 5. Multivariate Equation Systems -- 5.1. Multivariate Cryptography -- 5.1.1. Digital Signatures -- 5.1.2. The Oil and Vinegar Signature Scheme -- 5.1.3. Kipnis and Shamir's Cryptanalysis of the Oil and Vinegar Signature Scheme -- 5.2. Algebraic Cryptanalysis -- 6. The Advanced Encryption Standard -- 6.1. The Design of AES.

6.1.1. The AES S-box -- 6.1.2. Diffusion in AES -- 6.2. Geometric Properties of AES -- 6.2.1. The Group Generated by AES -- 6.2.2. The AES Difference Table -- 6.2.3. The BES Representation of AES -- 7. Concluding Remarks -- Acknowledgments -- References -- GALOIS GEOMETRIES AND LOW-DENSITY PARITY-CHECK CODES -- Abstract -- Introduction -- Constructions -- Structure of This Article -- 1. Low-Density Parity-Check Codes -- 2. Decoding of LDPC Codes -- 2.1. The Sum-product Algorithm -- 3. Assessing the Quality of an LDPC Code -- 4. Finite Incidence Structures and LDPC Codes -- 5. LDPC Codes from Linear Spaces -- 5.1. LDPC Codes Derived from Af ne Spaces -- 5.2. LDPC Codes Derived from Projective Spaces -- 5.3. Variations and Concluding Remarks -- 6. LDPC Codes from Partial Linear Spaces -- 6.1. LDPC Codes Derived from Generalized Quadrangles -- Further Results -- 6.2. LDPC Codes from Triangle-Free Geometries -- Further Constructions and Concluding Remarks -- 7. Open Problems -- Acknowledgm ents -- References -- Index.