On some invariants in numerical semigroups and estimations of the order bound

Abstract

Let S = {si}i∈IN ⊆ IN be a numerical semigroup. For si ∈ S, let ν(si) denote the number of pairs (si −sj, sj) ∈ S 2 . When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraicgeometric codes, a good bound for the minimum distance of the code Ci is the Feng and Rao order bound dORD(Ci) := min{ν(sj) : j ≥ i + 1}. It is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for si > sm, therefore dORD(Ci) = ν(si+1) for i ≥ m. By way of some suitable parameters related to the semigroup S, we find upper bounds for sm, we evaluate sm exactly in many cases, further we give a lower bound for several classes of semigroups. Index Therms. Numerical semigroup, Weierstrass semigroup, AG code, order bound on the minimum distance, Cohen-Macaulay type. ). e := s 1 > 1, the multiplicity of S. c := min {r ∈ S | r + IN ⊆ S}, the conductor of S d := the greatest element in S preceding c, the dominant of S c ′ := max{s i ∈ S | s i ≤ d and s i − 1 / ∈ S}, the subconductor of S d ′ := the greatest element in S preceding c ′ , when c ′ > 0 k := d − c ′ q := d − d ′ ℓ := c − 1 − d, the number of gaps of S greater than d g := #(IN \ S), the genus of S (= the number of gaps of S) τ := #(S(1) \ S), the Cohen−M acaulay type of S Since c − e − 1 / ∈ S we have c − e ≤ c ′ ; we define the following sets H := [c − e, c ′ ] ∩ IN \ S ⊆ IN \ S S ′ := {s ∈ S | s ≤ d ′ } ⊆ S.

Numerical Semigroups and Codes

“…In [48] Oneto and Tamone give further results on m and in [49] the same authors conjecture that for any numerical semigroup,…”

Ideals of Numerical Semigroups and Error-Correcting Codes

“…It is fundamental in the computation of bounds for the minimum distance of algebraic-geometry codes based on a single point as well as in the optimization of the redundancy of those codes. Its properties and applications can be seen in [7][8][9][10][11][12][13][14] and in the survey [1]. As a curiosity, it was proved in [7,15] that the set of elements of a numerical semigroup is determined by its ν sequence.…”

“…The minimum distance of the dual one-point code C ⊥ m associated with a rational point Q with Weierstrass semigroup Λ and associated sequence ν is bounded by the order (or Feng-Rao) bound defined as δ(m) = min{ν i : i > m} [3,10,47]. Some results about the computation of the order bound can be found in [3,7,[11][12][13][14]48].…”

New Lower Bounds on the Generalized Hamming Weights of AG Codes

“…and ν i = #D(i) for i ∈ N 0 . Some results related to the sequence ν i and also to its applications to coding theory can be found for instance in [25], [3], [4], [29], [31], [32], [33]. Barucci [2] proved the next result.…”

“…In particular, for an algebraic curve with Weierstrass semigroup Λ at a rational point P , the order (or Feng-Rao) bound on the minimum distance of the duals of the one-point codes defined on P by the evaluation of rational functions having only poles at P of order at most λ m is defined as δ(m) = min{ν i : i > m} [15], [25], [23]. Some results on its computation can be found in [8], [23], [3], [29], [31], [32], [33].…”