AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

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 $${\mathcal {X}}$$ X were investigated and the sets of minimal generators were determined for all points in $${\mathcal {X}}(\mathbb {F}_{q^2})$$ X ( F q 2 ) and $${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$ X ( F q 6 ) \ X ( F q 2 ) . This paper completes their work by settling the remaining cases, that is, for points in $${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$ 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 $${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$ X ( F q 7 ) \ X ( F q ) and we give a bound on the Feng–Rao minimum distance $$d_{ORD}$$ d ORD . For $$q=3$$ q = 3 we provide a table that also reports the exact values of $$d_{ORD}$$ d ORD . As a further application we construct quantum codes from $$\mathbb {F}_{q^7}$$ F q 7 -rational points of the GK-curve.

On reducible non-Weierstrass semigroups

Weierstrass semigroups are well known along the literature. We present a new family of non-Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired.

Ideals of Numerical Semigroups and Error-Correcting Codes

Several results relating additive ideals of numerical semigroups and algebraic-geometrycodes are presented. In particular, we deal with the set of non-redundant parity-checks, the codelength, the generalized Hamming weights, and the isometry-dual sequences of algebraic-geometrycodes from the perspective of the related Weierstrass semigroups. These results are related tocryptographic problems such as the wire-tap channel, t-resilient functions, list-decoding, networkcoding, and ramp secret sharing schemes.