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

TL;DR

This paper explicitly computes the second Feng-Rao number for inductive numerical semigroups, revealing its role in the asymptotic behavior of bounds for algebraic geometry codes, with applications to towers of function fields.

Contribution

It provides a new explicit formula for the second Feng-Rao number of inductive semigroups and explores properties and computational methods for these semigroups.

Findings

Explicit computation of the second Feng-Rao number for inductive semigroups.

Application of results to asymptotically good towers of function fields.

Development of tests to identify inductive numerical semigroups.

Abstract

The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behaviour of the order bound for the second Hamming weight of one-point AG codes. In particular, this result is applied for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, some properties of inductive numerical semigroups are studied, the involved Ap\'{e}ry sets are computed in a recursive way, and some tests to check whether a given numerical semigroups is inductive or not are provided.