On some invariants in numerical semigroups and estimations of the order bound

TL;DR

This paper investigates invariants of numerical semigroups, especially in the context of algebraic geometry codes, proposing bounds on the order and providing proofs for specific cases.

Contribution

It introduces new bounds for the Feng-Rao order in the context of Weierstrass semigroups and conjectures a universal inequality relating semigroup parameters and the order bound.

Findings

Proved the inequality in several cases.

Evaluated the Feng-Rao order bound for specific semigroups.

Conjectured a universal lower bound for the order bound.

Abstract

We study suitable parameters and relations in a numerical semigroup S. When S is the Weierstrass semigroup at a rational point P of a projective curve C, we evaluate the Feng-Rao order bound of the associated family of Goppa codes. Further we conjecture that the order bound is always greater than a fixed value easily deduced from the parameters of the semigroup: we also prove this inequality in several cases.