Acute Semigroups, the Order Bound on the Minimum Distance and the Feng-Rao Improvements
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We introduce a new class of numerical semigroups, which we call the class of 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 Λ={λ_0<λ_1<…} denote ν_i=#{j|λ_i-λ_j∈Λ}. Given an acute numerical semigroup Λ 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 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 non-decreasing are ordinary numerical semigroups. Furthermore we show that a semigroup can be uniquely determined by its sequence (ν_i).
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