On the order bound of one-point algebraic geometry codes

Let S={si}i∈N⊆NS={si}i∈N⊆N be a numerical semigroup. For each i∈Ni∈N, let ν(si)ν(si) denote the number of pairs (si−sj,sj)∈S2(si−sj,sj)∈S2: it is well-known that there exists an integer mm such that the sequence {ν(si)}i∈N{ν(si)}i∈N is non-decreasing for i>mi>m. The problem of finding mm is solved only in special cases. By way of a suitable parameter tt, we improve the known bounds for mm and in several cases we determine m explicitly. In particular we give the value of m   when the Cohen–Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When SS is the Weierstrass semigroup of a family {Ci}i∈N{Ci}i∈N of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {Ci}{Ci}.