On the order bound of one-point algebraic geometry codes

Abstract

Communicated by A.V. Geramita MSC:20M14 94B35 a b s t r a c t Let S = {s i } i∈N ⊆ N be a numerical semigroup. For each i ∈ N, let ν(s i ) denote the number of pairs (s i − s j , s j ) ∈ S 2 : it is well-known that there exists an integer m such that the sequence {ν(s i )} i∈N is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen-Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When S is the Weierstrass semigroup of a family {C i } i∈N of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C i }.

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

“…Moreover in every considered case we show that s m ≥ c + d − e. In Section 2, we establish various formulas and inequalities among the integers e, ℓ, d ′ , c ′ , d, c and t := d − s, see in particular (2.5) and (2.6). In Section 3, by using the results of Section 2 and some result from [6], we improve the known facts on s m recalled above; further we state the conjecture that c + d − e ia always a lower bound for s m and we prove it in many cases. Finally (Section 4) we deduce some particular case by applying the previous results and by some direct trick.…”

“…Case B. We have: 3, 5, 6, 7, 8, 9, 11} ⊆ Σ, 4 / ∈ Σ 2d − 9 (0, 9) (3,6) (c, 13) 2d − 10 (0, 10)(3, 7) (5,5) (c, 14) s m = 2d − 10 ⇐⇒ 10 ∈ Σ, 13 / ∈ Σ( =⇒ 9 ≤ t ≤ 10) otherwise either (α) 10, 13 ∈ Σ or (β) 10 / ∈ Σ (t = 7)…”

“…When S is the Weierstrass semigroup of a family {C i } i∈IN of one-point AG codes (see [3], [2]), a good bound for the minimum distance of C i is the Feng and Rao order bound d ORD (C i ) := min{ν(s j ) : j ≥ i + 1} where, for s j ∈ S, ν(s j ) denotes the number of pairs (s j −s k , s k ) ∈ S 2 (see [2]). It is well-known that there exists an integer m such that sequence {ν(s i )} i∈IN is non-decreasing for i ≥ m + 1 (see [7]) and so d ORD (C i ) = ν(s i+1 ) for i ≥ m. For this reason it is important to find the element s m of S. In our papers [5] and [6], we proved that s m = s + d if s ≥ d ′ , further we evaluated s m for ℓ ≤ 2, e ≤ 6, Cohen-M acaulay type ≤ 3.…”

“…We saw in [6], that s m = s + d, when s ≥ d ′ . To give estimations of s m in the remaining cases we shall use the same tools as in [6]: we recall them for the convenience of the reader and we add some improvement, as the general inequalities (…”

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.…”