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We construct examples of curves defined over the finite field F q 6 which are covered by the GK-curve. Thus such curves are maximal over F q 6 although they cannot be covered by the Hermitian curve for q > 2. We also give examples of maximal curves that cannot be Galois covered by the Hermitian curve over the finite field F q 2n with n > 3 odd and q > 2. We… (More)
We show that a maximal curve over F q 2 defined by the affine equation y n = f (x), where f (x) ∈ F q 2 [x] has degree coprime to n, is such that n is a divisor of q + 1 if and only if f (x) has a root in F q 2. In this case, all the roots of f (x) belong to F q 2 ; cf. we characterize certain maximal curves defined by equations of type y n = x m + x over… (More)
We characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely x m y a + y n + x b = 0. We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type y q + y = x m via class field theory.