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

@article{Oneto2010OnSI,
title={On some invariants in numerical semigroups and estimations of the order bound},
author={Anna Oneto and Grazia Tamone},
journal={Semigroup Forum},
year={2010},
volume={81},
pages={483-509}
}
• Published 9 February 2010
• Mathematics
• Semigroup Forum
Let S={si}i∈ℕ⊆ℕ be a numerical semigroup. For si∈S, let ν(si) denote the number of pairs (si−sj,sj)∈S2. When S is the Weierstrass semigroup of a family $\{\mathcal{C}_{i}\}_{i\in\mathbb{N}}$ of one-point algebraic-geometric codes, a good bound for the minimum distance of the code $\mathcal{C}_{i}$ is the Feng and Rao order bounddORD(Ci). It is well-known that there exists an integer m such that dORD(Ci)=ν(si+1) for each i≥m. By way of some suitable parameters related to the semigroup S, we find…
7 Citations
Smoothability and order bound for AS semigroups
• Mathematics
• 2012
In this paper we consider numerical semigroups S generated by arithmetic sequences m0,…,mn (AS-semigroups). First we state some results on the module $T^{1}_{k[S]}$; further in the cases m0≡1 and
Numerical Semigroups and Codes
A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N, given by the Weierstrass semigroup at one point of a curve.
New Lower Bounds on the Generalized Hamming Weights of AG Codes
• Computer Science
IEEE Transactions on Information Theory
• 2014
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized and used to bound the generalized Hamming weights of algebraic-geometry codes.
Ideals of Numerical Semigroups and Error-Correcting Codes
This work deals 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.
Deformations and smoothability of certain AS monomial curves
• Mathematics
• 2014
In this paper, we consider semigroups of embedding dimension five generated by arithmetic sequences. We prove these semigroups are Weierstrass by showing that the associated monomial curves are
Collected results on semigroups, graphs and codes
This thesis presents a compendium of _ve works where discrete mathematics play a key role, describing di_erent developments and applications of the semigroup theory while the other two have more independent topics.
On the minimum distance of AG codes, on Weierstrass semigroups and the smoothability of certain monomial curves in 4-Space
• Mathematics, Computer Science
• 2011
It is shown how to evaluate the Feng-Rao Order Bound, and it is proved that the semigroups of embedding dimension four generated by an arithmetic sequence are Weierstrass.

## References

SHOWING 1-8 OF 8 REFERENCES
Acute semigroups, the order bound on the minimum distance, and the Feng-Rao improvements
• M. Bras-Amorós
• Mathematics, Computer Science
IEEE Transactions on Information Theory
• 2004
It is shown that a semigroup can be uniquely determined by its sequence (/spl nu//sub i/), and it is proved that the only numerical semigroups for which the sequence (/ spl nu// Sub i/) is always nondecreasing are ordinary numericalSemigroups.
Codes from the Suzuki function field
This work determines and exploits the structure of the Weierstrass gap set of an arbitrary pair of rational places of F(2/sup 2n+1/)(x,y)/F( 2/Sup 2n-1/), and finds some codes over F/sub 8/ with parameters that are better than any known code.
A simple approach for construction of algebraic-geometric codes from affine plane curves
• Computer Science
IEEE Trans. Inf. Theory
• 1994
A novel approach for construction of AG codes without any background in algebraic geometry is presented, which indicates that the codes constructed by this approach are better than the current AG codes from same curves.
Minimal set of generators for the derivation module of certain monomial curves
• Mathematics
• 1999
Let Kbe a field of characteristic zero and let be a sequence of positive integers. Let C be an algebroid monomial curve in the affine e-space defined parametrically by Tme-1 and let A be the
The minimum distance of codes in an array coming from telescopic semigroups
• Computer Science
IEEE Trans. Inf. Theory
• 1995
The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound and is explained in terms of linear algebra and the theory of semigroups only.