On numerical semigroups and the order bound

Abstract

a b s t r a c tLet S = {s 0 = 0 < s 1 < · · · < s i . . .} ⊆ N be a numerical non-ordinary semigroup; then set, for each i, ν i := #{(s i − s j , s j ) ∈ S 2 }. We find a non-negative integer m such thatwhere 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, the value of m is also found.

On the order bound of one-point algebraic geometry codes

“…3.8] (2) ν(s i+1 ) ≥ ν(s i ), for every s i ≥ 2d + 1. [5,Prop. 3.9.1] (3) If S is an ordinary semigroup, then m = 0.…”

“…The meaning of t will be clear in the next Theorem 2.8 where we gather the known results on this argument (see [5,Th. 3.1]).…”

“…3.1]). We state beforehand a proposition which allows us to re-write in a better way the above cited Theorem 3.1 of [5]. In…”

Numerical Semigroups and Codes

“…Next we give the definition of near-acute semigroups. Oneto and Tamone in [47] proved that m = min{c + c ′ − 2 − g, 2d − g} if and only if c + c ′ − 2 2d or 2d − c + 1 ∈ Λ. Let us see next that these conditions in Oneto and Tamone's result are equivalent to having a near-acute semigroup.…”

“…Munuera and Torres in [45] and Oneto and Tamone in [47] proved that for any numerical semigroup m min{c + c ′ − 2 − g, 2d − g}. Notice that for acute semigroups this inequality is an equality.…”