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Rational points on curves over finite fields. Theory and Applications. (English) Zbl 0971.11033
London Mathematical Society Lecture Note Series. 285. Cambridge: Cambridge University Press. x, 245 p. £27.95; $ 39.95 (2001).

This monograph is a collection of the many contributions that the authors have made to the theory and applications of function fields over finite fields since 1995.

The first four chapters contain background material on function fields and class field theory, leading up to the authors’ applications of narrow ray class fields to produce function fields with many rational places, as in [Lect. Notes Comput. Sci. 1423, 555-566 (1998; Zbl 0909.11052)]. The fifth chapter contains the authors’ work on towers of global function fields with asymptotically many rational places, as in [Math. Nachr. 195, 171-186 (1998; Zbl 0920.11039)].

Chapter 6 contains applications to algebraic coding theory, and includes the recent constructions of new geometric codes using places of arbitrary degree, as developed by the authors and K. Y. Lam in [Appl. Algebra Eng. Commun. Comput. 9, 373-381 (1999; Zbl 1035.94016) and IEEE Trans. Inf. Theory 45, 2498-2501 (1999; Zbl 0956.94023)]. Chapter 7 contains applications to cryptography, including the construction of sequences with almost perfect linear complexity profile, which are used in stream ciphers; this work is due to the authors, K. Y. Lam and C. S. Ding [Finite Fields Appl. 5, 301-313 (1999; Zbl 0943.94005)]. The final chapter contains applications to low-discrepancy sequences, which are useful in quasi-Monte Carlo methods. The authors explain a construction of such sequences using function fields with many rational places, as in [Finite Fields Appl. 2, 241-273 (1996; Zbl 0893.11029)]. While the book deals almost exclusively with function fields, there is an appendix that discusses the connections between function fields and algebraic curves.

11G20Curves over finite and local fields
11-02Research monographs (number theory)
11R58Arithmetic theory of algebraic function fields
11R37Class field theory for global fields
14G05Rational points
94B27Geometric methods in coding theory
11K45Pseudo-random numbers; Monte Carlo methods
14H25Arithmetic ground fields (curves)
14G15Finite ground fields