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

Abstract

Let S={s i }i∈ℕ⊆ℕ be a numerical semigroup. For s i ∈S, let ν(s i ) denote the number of pairs (s i −s j ,s j )∈S 2. 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 bound d ORD (C i ). It is well-known that there exists an integer m such that d ORD (C i )=ν(s i+1) for each i≥m. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.