The number of points on a curve, and applications Arcs and curves: the legacy of Beniamino Segre

J. W. P. Hirschfeld
2006 Rendiconti di Matematica e delle sue Applicazioni. Serie VII  
Curves defined over a finite field have various applications, such as (a) the construction of good error-correcting codes, (b) the correspondence with arcs in a finite Desarguesian plane, (c) the Main Conjecture for maximum-distance-separable (MDS) codes. Bounds for the number of points of such a curve imply results in these cases. For plane curves, there is a variety of bounds that can be considered, such as the Hasse–Weil bound (1934/1948), the St¨ohr–Voloch bound (1986), as well as bounds
more » ... t depend on the plane embedding. Curves that achieve these bounds can sometimes be characterised. Segre applied bounds for the number of points on a curve to obtain bounds on the sizes of complete arcs. He also considered plane Fermat curves that achieve the Hasse–Weil bound. Various of these results and their applications are surveyed.
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