Niederreiter, Harald; Xing, Chaoping Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places. (English) Zbl 0891.11057 Acta Arith. 79, No. 1, 59-76 (1997). Let \(K\) be a global field with positive characteristic, that is the rational function field of a projective smooth algebraic curve over a finite field with \(q\) elements. It is of use for geometric coding theory [H. Stichtenoth, Algebraic function fields and codes (Springer, Berlin, 1993; Zbl 0816.14011)] and for construction of low-discrepancy sequences [H. Niederreiter and C. Xing, Finite Fields Appl. 2, No. 3, 241–273 (1996; Zbl 0893.11029)]] to describe explicitly such fields having the maximum (or at least, a number of places of degree 1 as close as possible to the) number \(N_q(g)\) of places of degree 1 for such fields of given genus \(g\).In the present paper, the authors give explicit examples of such fields over the finites fields with 3, 4 and 5 elements, for any genus less than or equal to 15, 15 (with a hole for genus 14) and 12 respectively. Such fields are given either in the form \(K=\mathbb{F}_{3,4\text{ or }5}(x,y_1,y_2)\) (where \(y_i\) is an algebraic expression of \(x\) given either by a Kummer or an Artin-Schreier extension), either as a Hayes cyclotomic extension of \(\mathbb{F}_{3,4\text{ or }5}(x)\).Note that other papers of the same authors give similar examples for binary, ternary, and quinary fields, in [Acta Arith. 75, 383–396 (1996; Zbl 0877.11065); in: Number theory (Ed. Győry, Pethő, Sos), de Gruyter (1998; Zbl 0923.11093); , Acta Arith. 83, No. 1, 65–86 (1998; Zbl 1102.11312); Demonstratio Math. 30, No. 4, 919–930 (1997; Zbl 0958.11075)]. Reviewer: Marc Perret (Lyon) Cited in 5 ReviewsCited in 6 Documents MSC: 11R58 Arithmetic theory of algebraic function fields 11R37 Class field theory Keywords:cyclotomic function fields; hilbert class fields; rational points; algebraic curves; geometric coding theory; construction of low-discrepancy sequences; global function fields; rational places Citations:Zbl 0816.14011; Zbl 0877.11065; Zbl 0893.11029; Zbl 0923.11093; Zbl 1102.11312; Zbl 0958.11075 PDFBibTeX XMLCite \textit{H. Niederreiter} and \textit{C. Xing}, Acta Arith. 79, No. 1, 59--76 (1997; Zbl 0891.11057) Full Text: DOI EuDML