Let S=$ \{s_i\}_{i\in\natsmall}\subseteq \nat$ be a numerical semigroup. For $s_i\in S$, let $\nu(s _ i)$ denote the number of pairs $ (s_i-s_j,s_j)\in S^2 $. When S is the Weierstrass semigroup of a family $\{C_i\}_{i\in{\mathbb N}}$ of one-point algebraic-geometric codes, a good bound for the minimum distance of the code $C_i$ is the Feng and Rao order bound $d_{ORD}( C_i) $. It is well-known that there exists an integer $m$ such that $d_{ORD}( C_i) =\nu(s_{i+1})$ for each $i\geq m$. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.
On some invariants in numerical semigroups and estimations of the order bound
ONETO, ANNA;TAMONE, GRAZIA
2010-01-01
Abstract
Let S=$ \{s_i\}_{i\in\natsmall}\subseteq \nat$ be a numerical semigroup. For $s_i\in S$, let $\nu(s _ i)$ denote the number of pairs $ (s_i-s_j,s_j)\in S^2 $. When S is the Weierstrass semigroup of a family $\{C_i\}_{i\in{\mathbb N}}$ of one-point algebraic-geometric codes, a good bound for the minimum distance of the code $C_i$ is the Feng and Rao order bound $d_{ORD}( C_i) $. It is well-known that there exists an integer $m$ such that $d_{ORD}( C_i) =\nu(s_{i+1})$ for each $i\geq m$. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.
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