Smoothability and order bound for AS semigroups.
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.
order bounds for algebraic-geometric codes
semigroups generated by arithmetic sequences
scientific article; zbMATH DE number 6136736
scientific article