The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field Fq arises naturally from the classical Dickson invariant of the projective linear group P GL(3, Fq). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q(q − 1)(q 3 − 2q − 2) + 1. In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse-Witt invariant is positive; the Fermat curve of degree q − 1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over F q 3 , the DGZ curve is optimal with respect the number of its F q 3 -rational points.
We investigate several types of linear codes constructed from two familiesS q and R q of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup H(P ) at an F q -rational point P is shown to be symmetric.
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