For each integer s ≥ 1, we present a family of curves that are $𝔽_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $𝔽_q$-Frobenius nonclassical with respect to the linear system of conics. In the $𝔽_q$-Frobenius nonclassical cases, we determine the exact number of $𝔽_q$-rational points. In the remaining cases, an upper bound for the number of $𝔽_q$-rational points will follow from Stöhr-Voloch theory.
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We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves, which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.
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