Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

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Analytic Number Theory

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

Introduction to finite fields and their applications

This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference.

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A course in combinatorics

Finite Generalized Quadrangles

On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

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The Geometry of Two-Weight Codes

This paper examines subsets with at most n points on a line in the projective plane π q = PG(2, q) A lower bound for the size of complete (k, n)-arcs is established and shown to be a generalisation of a classical result by Barlotti A sufficient condition ensuring that the trisecants to a complete (k, 3)-arc form a blocking set B in the dual plane π∗ q is provided Finally, combinatorial arguments are used to show that, for q ≥ 17, plane (k, 3)-arcs satisfying a prescribed incidence condition do not attain the best known upper bound

Bounds for Arcs of Arbitrary Degree in Finite Desarguesian Planes

The chords of a twisted cubic in PG(3, q) are mapped via their Plucker coordinates to the points of a Veronese surface lying on the Klein quadric in PG(5, q). This correspondence over a finite field gives a cap in PG(5, q), that is, a set of points no three of which are collinear. The dual structure, namely the axes of the osculating developable, is also mapped to a Veronese surface. The two surfaces can be combined to give a larger cap. The constructions can be extended to the chords and axes of an arbitrary (q+ I)-arc in PG(3, q) when q is even. An alternative construction for the cap associated to a twisted cubic is given for q odd. Research supported by G.N.S.A.G.A. of C.N.R. and the Italian Ministry for Research and Technology. The third author is a senior research assistant of the Belgian National Fund for Scientific Research. Australasian Journal of Combinatorics 16(1997). pp.99-111

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Applications of line geometry, III: The quadric Veronesean and the chords of a twisted cubic.

The dimension of projective geometry codes

The automorphism group of the set of 12 points associated with an apolar system of conics is determined. A complete (q − 1)-arc for q = 13 can be obtained as a special case. The orbits of its automorphism group are also described. © 1994 John Wiley & Sons, Inc.

The automorphism group of a complete (q − 1)‐arc in PG(2,q)

Publisher Summary This chapter discusses the caps in elliptic quadrics. In PG(n,q), projective space of n dimensions over the Galois field GF(q), a k-cap is a set of k points no three of which are collinear. The maximum value that k can take is denoted by m 2 (n,q). When n = 2, a k-cap is usually called a k-arc. For q odd, a (q+l)-arc is a conic and a (q 2 +1)-cap is an elliptic quadric. The three conics of a tetrahedral system are contained in a quadric if and only if the system is flat: the quadric is then unique and irreducible.

J. W. P. Hirschfeld

Papers

Bounds for Arcs of Arbitrary Degree in Finite Desarguesian Planes

Applications of line geometry, III: The quadric Veronesean and the chords of a twisted cubic.

The dimension of projective geometry codes

The automorphism group of a complete (q − 1)‐arc in PG(2,q)

Caps in Elliptic Quadrics