On the minimum distance of AG codes, on Weierstrass semigroups and the smoothability of certain monomial curves in 4-Space

摘要

In this paper we treat several topics regarding numerical Weierstrasssemigroups and the theory of Algebraic Geometric Codes associated to a pair$(X, P)$, where $X$ is a projective curve defined over the algebraic closure ofthe finite field $F_q$ and P is a $F_q$-rational point of $X$. First we showhow to evaluate the Feng-Rao Order Bound, which is a good estimation for theminimum distance of such codes. This bound is related to the classicalWeierstrass semigroup of the curve $X$ at $P$. Further we focus our attentionon the question to recognize the Weierstrass semigroups over fields ofcharacteristic 0. After surveying the main tools (deformations andsmoothability of monomial curves) we prove that the semigroups of embeddingdimension four generated by an arithmetic sequence are Weierstrass.