The root diagram for one-point AG codes arising from certain curves with separated variables

Year 2018, Volume: 5 Issue: 2, 71 - 83, 15.05.2018

Federico Fornasiero Guilherme Tizziotti

https://doi.org/10.13069/jacodesmath.423733

Abstract

Heegard, Little and Saints introduced in [8] an encoding algorithm for a class of AG codes via Gröbner
basis more compact compared with the usual encoding via generator matrix. So, knowing that the
main drawback of Gröbner basis is the high computational cost required for its calculation, in [12],
the same authors introduced the concept of root diagram that allows the construction of an algorithm
for computing a Gröbner basis with a lower complexity for one-point Hermitian codes. In [4], Farrán,
Munuera, Tizziotti and Torres extended the results obtained in [12] for codes on norm-trace curves.
In this work we generalize these results by constructing the root diagram for codes arising from certain
curves with separated variables that has certain special automorphism and a Weierstrass semigroup
generated by two elements. Such family of curves includes the norm-trace curve, among other curves
with recent applications in coding theory.

Keywords

AG codes, Gröbner basis, Root diagram

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