On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators

Abstract

Abstract. The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behaviour of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r = 1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower-bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.

On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

“…The results in [FG] improve previous bounds of Pellikaan in [KP] or the Griesmer order bound (see [HKM] and [DFGL2]), so that the results of this paper also improve them, as a consequence. More precisely, Pellikaan bound in [KP,Theorem 2.8] for r = 2 states that…”

New Lower Bounds on the Generalized Hamming Weights of AG Codes

“…Now, substituting N by its value, we check that n ℓ = q ⌊ ⌊m+1−log q (ℓ+1)⌋ 2 ⌋ − 1. The floor in the numerator of the exponent is redundant, and so this result coincides with (12).…”

“…The Weierstrass semigroup at the rational point at infinity is generated by q and q + 1 [38], [23]. Some results concerning this semigroup can be found in [6] and, in particular, concerning the weight hierarchy, in [12].…”

Isolated factorizations and their applications in simplicial affine semigroups

“…These applications come from the connection between the minimal presentations and the Betti elements of a monoid and are particularly interesting in the case of complete intersection affine semigroups, whose definition is recalled in Section 2.2. Complete intersection semigroups are relevant outside the theory of numerical semigroups [2,19,12,9] and, consequently, they have been the main topic of several research papers. In the search of families of complete intersection semigroups, Bertin and Carbonne introduced in [3] the concept of free numerical semigroups, which allows to construct complete intersection numerical semigroups of any desired embedding dimension.…”