On the weight hierarchy of codes coming from semigroups with two generators

TL;DR

This paper computes Feng-Rao numbers for semigroups with two generators, providing a simple formula that improves bounds on the generalized Hamming weights of algebraic geometry codes.

Contribution

It derives a closed-form formula for Feng-Rao numbers for two-generator semigroups using Apéry sets, enhancing bounds on code weights.

Findings

Derived a simple formula for Feng-Rao numbers in two-generator semigroups.

Improved bounds on generalized Hamming weights of algebraic geometry codes.

Compared and improved classical bounds like the Griesmer bound.

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 > 1 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 Ap\'ery 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.