Let S=$\{$ s$_0=0 < $s_1 < ... < s_i ... \}\subseteq {\mathbb N}$ be a numerical non-ordinary semigroup; then set, for each i, $\nu _ i := cardinality $ \{ (s _j, s_i-s_j)\in S ^2 \}$. \ We find a non-negative integer m such that $d_{ORD} (i)= \nu_{i +1}$ for $i\geq m$, where $d_{ORD} (i)$ denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, it is also found the value of m.
On numerical semigroups and the order bound
ONETO, ANNA;TAMONE, GRAZIA
2008-01-01
Abstract
Let S=$\{$ s$_0=0 < $s_1 < ... < s_i ... \}\subseteq {\mathbb N}$ be a numerical non-ordinary semigroup; then set, for each i, $\nu _ i := cardinality $ \{ (s _j, s_i-s_j)\in S ^2 \}$. \ We find a non-negative integer m such that $d_{ORD} (i)= \nu_{i +1}$ for $i\geq m$, where $d_{ORD} (i)$ denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, it is also found the value of m.
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