The second Feng-Rao number for codes coming from telescopic semigroups

TL;DR

This paper proves that for telescopic numerical semigroups, the second Feng-Rao number equals the semigroup's multiplicity, and explores implications for bounds on algebraic geometry codes.

Contribution

It establishes a precise relationship between the second Feng-Rao number and the multiplicity for telescopic semigroups, and analyzes Apéry sets under gluings.

Findings

Second Feng-Rao number equals the multiplicity in telescopic semigroups

Provides bounds for the second Hamming weight of algebraic geometry codes

Improves existing estimates like the Griesmer Order Bound

Abstract

In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Ap\'ery sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound.