On the order bound of one-point algebraic geometry codes

Abstract

Let S ={si}i∈IN ⊆ IN be a numerical semigroup. For each i ∈ IN, let ν(si) denote the number of pairs (si−sj, sj) ∈ S 2 : it is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m 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 lower or equal to six. When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {Ci}. Index Terms. Numerical semigroup, Weierstrass semigroup, algebraic geometry code, order bound on the minimum distance