The second Feng–Rao number for codes coming from telescopic semigroups

Abstract

In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Apéry sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound.

Isolated factorizations and their applications in simplicial affine semigroups

“…These applications come from the connection between the minimal presentations and the Betti elements of a monoid and are particularly interesting in the case of complete intersection affine semigroups, whose definition is recalled in Section 2.2. Complete intersection semigroups are relevant outside the theory of numerical semigroups [2,19,12,9] and, consequently, they have been the main topic of several research papers. In the search of families of complete intersection semigroups, Bertin and Carbonne introduced in [3] the concept of free numerical semigroups, which allows to construct complete intersection numerical semigroups of any desired embedding dimension.…”

On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

“…For any k ≤ j − 1, we have ρ k < ρ k + x < ρ k + d k = ρ k+1 , and so ρ k − (−x) = ρ k + x / ∈ Γ. By Lemma 8, this implies that Ap(Γ, −d j ) ⊆ Ap(Γ, −x), and thus by [FGHL,Lemma 1],…”

“…Proof. We only need to use the fact that #Ap(Γ, x) = #Ap(Γ, −x) + x (see [FGHL,Lemma 1]) together with Lemma 8.…”

Ideals of Numerical Semigroups and Error-Correcting Codes

“…Farrán and Munuera showed the existence of a constant, which they named the Feng-Rao number, depending only on the dimension of the Hamming weights and the Weierstrass semigroup, which completely determined the order bounds for codes of rate low enough. The references [41][42][43][44] deal with the generalized order bounds and the Feng-Rao numbers related to particular classes of semigroups.…”