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In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve X were investigated and the sets of minimal generators were determined for all points in X (F q 2) and X (F q 6) \ X (F q 2) . This paper completes their work by settling the remaining cases, that is, for points in X (F ¯ q) \ X (F q 6) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in X (F q 7) \ X (F q) and we give a bound on the Feng–Rao minimum distance d ORD . For q = 3 we provide a table that also reports the exact values of d ORD . As a further application we construct quantum codes from F q 7 -rational points of the GK-curve. [ABSTRACT FROM AUTHOR]

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman's 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192. [ABSTRACT FROM AUTHOR]