Abstract
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 {mathcal {X}} were investigated and the sets of minimal generators were determined for all points in {mathcal {X}}(mathbb {F}_{q^2}) and {mathcal {X}}(mathbb {F}_{q^6})setminus {mathcal {X}}( mathbb {F}_{q^2}). This paper completes their work by settling the remaining cases, that is, for points in {mathcal {X}}(overline{mathbb {F}}_{q}){setminus }{mathcal {X}}( mathbb {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 {mathcal {X}}(mathbb {F}_{q^7}){setminus }{mathcal {X}}( mathbb {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 mathbb {F}_{q^7}-rational points of the GK-curve.
Highlights
Algebraic geometric methods have largely been used for the construction of error-correcting linear codes from algebraic curves
The essential idea going back to Goppa’s work is that a linear code can be obtained from an algebraic curve X defined over a finite field q by evaluating certain rational functions whose poles are prescribed by a given q-rational divisor G at some q-rational divisor D whose support is disjoint from that of G
AG codes from q7‐rational points of the GK maximal curve. This theorem together with the already quoted previous results provide a complete description of the Weierstrass semigroups at any point of the GK-curve
Summary
Algebraic geometric methods have largely been used for the construction of error-correcting linear codes from algebraic curves. The essential idea going back to Goppa’s work (see [10] and [11]) is that a linear code can be obtained from an algebraic curve X defined over a finite field q by evaluating certain rational functions whose poles are prescribed by a given q-rational divisor G at some q-rational divisor D whose support is disjoint from that of G. For q = 2, the minimal set of generators for H(P) is {7, 8, 12, 13, 18} This theorem together with the already quoted previous results provide a complete description of the Weierstrass semigroups at any point of the GK-curve. = 5, The above results are applied to the construction of AG codes and quantum codes from an q7-rational point of the GK curve.
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