$B_{h}[g]$ modular sets from $B_{h}$ modular sets

TL;DR

This paper explores the construction of $B_{h}[g]$ modular sets from existing $B_{h}$ modular sets using homomorphisms, analyzing classical constructions to identify methods that preserve set cardinality.

Contribution

It introduces a new approach to construct $B_{h}[g]$ modular sets from $B_{h}$ sets via homomorphisms, extending classical methods.

Findings

Constructed $B_{h}[g]$ sets from $B_{h}$ sets using homomorphisms.

Analyzed classical $B_{h}$ set constructions for preserving set size.

Provided conditions under which the cardinality of sets is maintained.

Abstract

A set of positive integers $A$ is called a $B_{h}[g]$ set if there are at most $g$ different sums of $h$ elements from $A$ with the same result. This definition has a generalization to abelian groups and the main problem related to this kind of sets, is to find $B_{h}[g]$ maximal sets i.e. those with larger cardinality. We construct $B_{h}[g]$ modular sets from $B_{h}$ modular sets using homomorphisms and analyze the constructions of $B_{h}$ sets by Bose and Chowla, Ruzsa, and G\'omez and Trujillo look at for the suitable homomorphism that allows us to preserve the cardinal of this types of sets.