J. W. P. Hirschfeld

Bio: J. W. P. Hirschfeld is an academic researcher from University of Sussex. The author has contributed to research in topics: Finite geometry & Projective space. The author has an hindex of 23, co-authored 56 publications receiving 4582 citations. Previous affiliations of J. W. P. Hirschfeld include University of Brighton.

Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference

Abstract: This is a graduate-level book aimed at experienced researchers as well as beginning graduate students in finite geometry, incidence geometry, design theory, coding theory and combinatorics. The 21 articles arise from the Fourth Isle of Thorns Conference on Finite Geometries, held in July 2000. The book contains new high-level results at the forefront of the field as well as authoritative surveys. The articles are written so that beginning graduate students can appreciate them. Among the surveys by the invited speakers are P.J. Cameron `Fixed Points and cycles', C.E. Praeger `Implications of line-transitivity for designs', B. Schmidt, `Exponent bounds', which explains recent results on difference sets, and several papers with H. Van Maideghem as co-author on generalized polygons. The editors also contribute: A. Blokhuis, D. Jungnickel, and B. Schmidt `On a class of symmetric divisible designs which are almost projective planes', and D. Luyckx and J.A. Thas discuss the geometry of the quadric in six dimensions. There is also a lengthy and complete survey by J.W.P. Hirschfeld and L. Storme `The packing problem in statistics, coding theory and finite projective spaces: update 2001'. Audience: Researchers as well as beginning graduate students in the areas of finite geometry, incidence geometry, design theory, coding theory and combinatorics will appreciate the surveys and original papers by leaders in these fields.

Families of curves covered by the Hermitian curve

Abstract: Let K be the finite field of order $q^2$. For every positive divisor m of q+1 for which d =(q+1)/m is prime, the plane curves C with the given affine equation are covered by the Hermitian curve. The non-singular K-models of these curves are K-maximal and provide examples of non-isomorphic curves with the same genus and the same automorphism group. The case m=2 was previously investigated by the authors.

Finite geometry after Aschbacher's Theorem: PGL(n, q) from a Kleinian viewpoint

Abstract: Studying the geometry of a group G leads us to questions about its maximal subgroups and primitive permutation representations (the G-invariant relations and similar structures, the base size, recognition problems, and so on). Taking the point of view that nite projective geometry is the geometry of the groups PGL(n; q), Aschbacher's theorem gives us eight natural families of geometric objects, with greater or smaller degrees of familiarity. This paper presents some speculations on how the subject could develop from this point of view.

The Desarguesian Plane of Order Thirteen

Abstract: The algebraic curve associated to an arc in PG(2, q), with q odd, is examined using both properties of the curve itself as well as properties of the arc. The key case of (q − 1)-arcs means that the behaviour of the associated sextic curves needs to be studied. The case of PG(2, 13) is examined in detail. There is a geometric bijection between 12-arcs and their duals. The latter lead to optimal sextic curves; the former lead to sextics whose set of rational points make them ‘look like’ quartics.

Analytic Number Theory

Abstract: 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.

Introduction to finite fields and their applications

Abstract: 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.

A course in combinatorics

Abstract: 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.

The Geometry of Two-Weight Codes

Abstract: 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