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

1 Abstract. Let S = {si}i∈IN ⊆ IN be a numerical semigroup. For si ∈ S, let ν(si) denote the number of pairs (si−sj , sj) ∈ S . When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraicgeometric codes, a good bound for the minimum distance of the code Ci is the Feng and Rao order bound dORD(Ci) := min{ν(sj) : j ≥ i+ 1}. It is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for si > sm, therefore dORD(Ci) = ν(si+1) for i ≥ m. By way of some suitable parameters related to the semigroup S, we find upper bounds for sm, we evaluate sm exactly in many cases, further we give a lower bound for several classes of semigroups.