We introduce a new class of numerical semigroups, which we call the class of { \it acute} semigroups and we prove that they generalize symmetric and pseudo-symmetric numerical semigroups, Arf numerical semigroups and the semigroups generated by an interval. For a numerical semigroup $\Lambda=\{ \lambda_0<\lambda_1<\dots\}$ denote $\nu_i=\#\{ j\mid\lambda_i-\lambda_j\in\Lambda\}$. Given an acute numerical semigroup $\Lambda$ we find the smallest non-negative integer $m$ for which the order bound on the minimum distance of one-point Goppa codes with associated semigroup $\Lambda$ satisfies $d_{ ORD}(C_i)(:=\min\{ \nu_j\mid j>i\})=\nu_{ i+1}$ for all $i\geq m$. We prove that the only numerical semigroups for which the sequence $(\nu_i)$ is always non-decreasing are ordinary numerical semigroups. Furthermore we show that a semigroup can be uniquely determined by its sequence $(\nu_i)$.
Categories: cs.IT cs.DM math.IT