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.

Ovoids, spreads and m-systems of finite classical polar spaces

Abstract: In this chapter, ovoids, spreads and m-systems of finite classical polar spaces are introduced Also SPG-reguli, SPG-systems, BLT-sets and sets with the BLT-property are defined The main results on these topics are given, all without proof

On plane maximal curves

Abstract: The number N of rational points on an algebraic curve of genus g over a finite field \({\mathbb{F}}_q \) satisfies the Hasse–Weil bound \(N \leqslant q + 1 + 1g\sqrt q \). A curve that attains this bound is called maximal. With \(g_0 = \frac{1}{2}(q - \sqrt q )\) and \(g_1 = \frac{1}{4}(\sqrt q - 1)^2 \), it is known that maximalcurves have \(g = g_0 or g \leqslant {\text{ }}g_1 \). Maximal curves with \(g = g_0 or g_1 \) have been characterized up to isomorphism. A natural genus to be studied is \(g_2 = \frac{1}{8}(\sqrt q - 1)(\sqrt q - 3),\) and for this genus there are two non-isomorphic maximal curves known when \(\sqrt q \equiv 3 (\bmod 4)\). Here, a maximal curve with genus g2 and a non-singular plane model is characterized as a Fermat curve of degree \(\frac{1}{2}(\sqrt q + 1)\).

Linear independence in finite spaces

Abstract: The maximum number m2(n, q) of points in PG(n, q), n⩾2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q n−1 −1/2qn−2 is obtained for q even when n⩾4 and q>2. A necessary preliminary is an improved upper bound for m′2(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that \(m'_2 (3,q){\text{ }} \leqslant q^2 - \tfrac{1}{2}{\text{q}} - \tfrac{1}{2}\sqrt {\text{q}} {\text{ + 2}}\) and that m′2(3, 4)=14.

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