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 $\Lambda =\{{\lambda }_0<{\lambda }_1<\dots \}$ 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 $i\ge m$. 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 $({\nu }_i)$.

Published in IEEE Trans. Inf. Theory, 2004. MR2094884. arXiv admin note: substantial text overlap with arXiv:1706.09765

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