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

TL;DR

This paper introduces acute semigroups, a new class that generalizes several known semigroups, and explores their properties related to the order bound on minimum distances in algebraic geometry codes.

Contribution

It defines acute semigroups, proves their relation to existing classes, and characterizes the sequence mbda_i, including conditions for non-decreasing sequences and unique determination of semigroups.

Findings

Acute semigroups generalize symmetric, pseudo-symmetric, Arf, and interval-generated semigroups.

The sequence (mbda_i) is non-decreasing only for ordinary semigroups.

Semigroups are uniquely determined by their sequence (mbda_i).

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 $\Lambda=\{\lambda_0<\lambda_1<\dots\}$ 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 $(\nu_i)$.