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Automorphism groups of Ree type, Deligne-Lusztig curves and function fields. (English) Zbl 0769.14009

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.

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
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