Finite Fields and Applications : Proceedings of The Fifth International Conference on Finite Fields and Applications F q 5, held at the University of Augsburg, Germany, August 2-6, 1999

The Fifth International Conference on Finite Fields and Applications F q5 held at the University of Augsburg, Germany, from August 2-6, 1999 continued a series of biennial international conferences on finite fields. The proceedings document the steadily increasing interest in this topic. Finite fields have an inherently fascinating structure and are important tools in discrete mathematics. Their applications range from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields. This interplay has always been a dominant theme in F q conferences and was very much in evidence at F q5. The proceedings reflect this, and offer an up-to-date collection of surveys and original research articles by leading experts in the area. Front Matter....Pages I-IX Projective Generalized Reed-Muller Codes over p -adic Numbers and Finite Rings....Pages 1-13 Towards a Basis for the Space of Regular Functions in a Tower of Function Fields Meeting the Drinfeld-Vladut Bound....Pages 14-24 Self-Dual Normal Bases and Related Topics....Pages 25-36 Permutations amongst the Dembowski-Ostrom Polynomials....Pages 37-42 Associative Rational Functions in Two Variables....Pages 43-56 A Note on the Minimal Polynomial of the Product of Linear Recurring Sequences....Pages 57-69 Gelfond-Gramain’s Theorem for Function Fields....Pages 70-80 On Generalized Bent and q -ary Perfect Nonlinear Functions....Pages 81-94 Divisible Designs Admitting, as an Automorphism Group, an Orthogonal Group or a Unitary Group....Pages 95-108 A Permutation of a Small Infinite Field....Pages 109-112 Almost Perfect Nonlinear Power Functions on GF(2 n ): A New Case for n Divisible by 5....Pages 113-121 On Non-Abelian Semi-Regular Relative Difference Sets....Pages 122-127 Applications of Arithmetical Geometry to Cryptographic Constructions....Pages 128-161 Gauß Periods in Finite Fields....Pages 162-177 Universal Generators for Primary Closures of Galois Fields....Pages 178-195 Irreducible Polynomials Generated by Decimations....Pages 196-207 Decoding Reed-Muller Codes beyond Half the Minimum Distance....Pages 208-223 On Binary Cyclic Codes With Few Weights....Pages 224-231 MacWilliams Identities for Linear Codes over Finite Frobenius Rings....Pages 232-250 Cyclotomic Function Fields with Many Rational Places....Pages 251-275 On the Siamese Twin Designs....Pages 276-292 On the Brauer Monoid for Finite Fields....Pages 293-302 Algorithms for Factoring Polynomials over Finite Fields....Pages 303-312 Gauss Sums over Quasi-Frobenius Rings....Pages 313-318 On the Early History of Galois Fields....Pages 319-328 Linear Blocking Sets: A Survey....Pages 329-340 Comparison of Techniques on Divisibility Properties of Exponential Sums and Applications....Pages 341-355 On the Rank of Appearance and the Number of Zeros of the Lucas Sequences over $$ {\mathbb{F}_{q}} $$ ....Pages 356-362 A Permutation Problem for Finite Fields....Pages 363-389 The L -Function of Gold Exponential Sum....Pages 390-408 On Permutation Polynomials and Derivable Translation Planes....Pages 409-417 Pure L -Functions from Algebraic Geometry over Finite Fields....Pages 418-427 Incomplete Additive Character Sums and Applications....Pages 428-436 Applications of Algebraic Curves to Constructions of Codes and Almost Perfect Sequences....Pages 437-461 Back Matter....Pages 462-474 ....Pages 475-489 Projective Generalized Reed-Muller Codes over p-adic Numbers and Finite Rings Towards a Basis for the Space of Regular Functions in a Tower of Function Fields Meeting the Drinfeld-Vladut Bound Self-Dual Normal Bases and Related Topics Permutations amongst the Dembowski-Ostrom Polynomials Associative Rational Functions in Two Variables A Note on the Minimal Polynomial of the Product of Linear Recurring Sequences Gelfond-Gramain's Theorem for Function Fields On Generalized Bent and q-ary Perfect Nonlinear Functions Divisible Designs Admitting, as an Automorphism Group, an Orthogonal Group or a Unitary Group. A Permutation of a Small Infinite Field Almost Perfect Nonlinear Power Functions on GF(2n): A New Case for n Divisible by 5 On Non-Abelian Semi-Regular Relative Difference Sets Applications of Arithmetical Geometry to Cryptographic Constructions Gauß Periods in Finite Fields Constructions of Orthomorphisms of?2n Period Polynomials for $$ {F_{{{p̂{2}}}}} $$ of Fixed Small Degree Universal Generators for Primary Closures of Galois Fields Irreducible Polynomials Generated by Decimations Decoding Reed-Muller Codes beyond Half the Minimum Distance On Binary Cyclic Codes With Few Weights Mac Williams Identities for Linear Codes over Finite Frobenius Rings Cyclotomic Function Fields with Many Rational Places On the Siamese Twin Designs On the Brauer Monoid for Finite Fields Algorithms for Factoring Polynomials over Finite Fields Gauss Sums over Quasi-Frobenius Rings On the Early History of Galois Fields Linear Blocking Sets: a Survey Comparison of Techniques on Divisibility Properties of Exponential Sums and Applications On the Rank of Appearance and the Number of Zeros of the Lucas Sequences over Fq A Permutation Problem for Finite Fields The L-Function of Gold Exponential Sum On Permutation Polynomials and Derivable Translation Planes Pure L-Functions from Algebraic Geometry over Finite Fields Incomplete Additive Character Sums and Applications Applications of Algebraic Curves to Constructions of Codes and Almost Perfect Sequences Author Index This volume represents the refereed proceedings of the Fifth International Conference on Finite Fields and Applications (F q5) held at the University of Augsburg (Germany) from August 2-6, 1999, and hosted by the Department of Mathematics. The conference continued a series of biennial international conferences on finite fields, following earlier conferences at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University ofGlasgow (Scotland) in July 1995, and the University ofWaterloo (Canada) in August 1997. The Organizing Committee of F q5 comprised Thomas Beth ( The proceedings of the 5th International Conference on Finite Fields and Applications look at the applications of finite fields, ranging from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing.