Smoothability and order bound for AS semigroups.
type: Item
instance of: scholarly article
title: Smoothability and order bound for AS semigroups.
zbMATH Open document ID: 1270.20063
author: Grazia Tamone
author: Anna Oneto
published in: Semigroup Forum
publication date: 2013-02-15T00:00:00Z
review text: The additive semigroup \(S\) generated by the set \(M=\{m_0,m_1,\dots,m_n\}\) of positive integers is called an AS-semigroup, if \((m_0,m_1,\dots,m_n)=1\) and \(M\) is an arithmetic progression forming a minimal set of generators of \(S\). The authors show that if \(m\bmod n\in\{0,1\}\), then \(S\) is a Weierstrass semigroup. Recall that \(S\) is a Weierstrass semigroup if there exists a smooth projective curve \(X\) defined over an algebraically closed field \(k\) and there exists a Weierstrass point \(Q\) on \(X\) (as defined by \textit{F. K. Schmidt} [in Math. Z. 45, 75-96 (1939; Zbl 0020.10202)]) such that \(S\) is the complement of the set \(\{n>0:\dim L((n-1)Q)=\dim L(nQ)\}\), where \(L(nQ)\) is the linear space \[ \{f\in k(X)\setminus 0:\text{div}(f)+nQ\geq 0\}\cup\{0\}. \] This is then applied to provide explicit values for the Feng-Rao order bound function [\textit{G.-L. Feng} and \textit{T. R. N. Rao}, IEEE Trans. Inf. Theory 39, No. 1, 37-45 (1993; Zbl 0765.94021)], of importance in the algebraic-geometric codes theory.
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 11D07
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 6136736
zbMATH Keywords: numerical semigroups
zbMATH Keywords: Weierstrass semigroups
zbMATH Keywords: order bounds for algebraic-geometric codes
zbMATH Keywords: semigroups generated by arithmetic sequences
zbMATH Keywords: minimal generating sets
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: s00233-012-9405-3
OpenAlex ID: W1996256648
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DOI: 10.1007/S00233-012-9405-3
description: scientific article; zbMATH DE number 6136736