Acute semigroups, the order bound on the minimum distance, and the Feng-Rao improvements

We introduce a new class of numerical semigroups, which we call the class of acute semigroups and we prove that they generalize symmetric and pseudosymmetric numerical semigroups, Arf numerical semigroups, and the semigroups generated by an interval. For a numerical semigroup Λ={λ₀<λ₁<...}, denote ν_i=#{j|λ_i-λ_j∈Λ}. Given an acute numerical semigroup Λ we find the smallest nonnegative integer m for which the order bound on the minimum distance of one-point Goppa codes with associated semigroup Λ satisfies d_ORD(C_i)(:=min{ν_j|j>i})=ν_i+1 for all i≥m. We prove that the only numerical semigroups for which the sequence (ν_i) is always nondecreasing are ordinary numerical semigroups. Furthermore, we show that a semigroup can be uniquely determined by its sequence (ν_i).