Hansen, Johan P.; Pedersen, Jens Peter Automorphism groups of Ree type, Deligne-Lusztig curves and function fields. (English) Zbl 0769.14009 J. Reine Angew. Math. 440, 99-109 (1993). This paper studies and proves uniqueness of the Deligne-Lusztig curves \(X\) and corresponding function fields constructed via the Ree groups over \(\mathbb{F}_ q\), where \(q\) is an odd power of 3.The curves are particular in two senses. First of all, the number of \(\mathbb{F}_ q\)-rational points on \(X\), being \(1+q^ 3\), is the largest a curve of genus \(g=g(X)={3\over 2}q_ 0(q-1)(q+q_ 0+1)\), \((q_ 0=3^ s)\) can have. Secondly, they have very large groups of automorphisms compared to \(g(X)\), their sizes \(q^ 3(q-1)(q^ 3+1)\) exceed by far the Hurwitz upper bound \(84(g-1)\) valid in characteristic zero.The curves and corresponding function fields are of interest in the theory of error correcting codes via the Goppa construction. It is possible to construct codes of length \(q^ 3\) over \(\mathbb{F}_ q\), such that dimension+minimum distance \(\geq 1+q^ 3-g\). The large groups of automorphisms equip the resulting geometric Goppa codes with correspondingly large symmetry. Reviewer: J.P.Hansen (Aarhus) Cited in 1 ReviewCited in 11 Documents MSC: 14H05 Algebraic functions and function fields in algebraic geometry 94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory 14H25 Arithmetic ground fields for curves 11G20 Curves over finite and local fields 14G15 Finite ground fields in algebraic geometry 11R58 Arithmetic theory of algebraic function fields Keywords:number of rational points; Deligne-Lusztig curves; function fields; large groups of automorphisms; Goppa codes PDFBibTeX XMLCite \textit{J. P. Hansen} and \textit{J. P. Pedersen}, J. Reine Angew. Math. 440, 99--109 (1993; Zbl 0769.14009) Full Text: Crelle EuDML