Acute Semigroups, the Order Bound on the Minimum Distance and the Feng-Rao Improvements

Abstract

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 $Λ=\{λ_0<λ_1<\dots\}$ denote $ν_i=\#\{j\midλ_i-λ_j\inΛ\}$. 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 $d_{ORD}(C_i)(:=\min\{ν_j\mid j>i\})=ν_{i+1}$ for all $i\geq m$. 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 $(ν_i)$.