Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory

TL;DR

This paper introduces and analyzes two new families of weighted zero-sum constants for finite abelian groups, establishing their properties, relationships, and connections to coding theory and combinatorial structures.

Contribution

It defines the m-wise and d-constrained Davenport constants with weights, links them, and derives explicit results for elementary p-groups, connecting to coding theory and cap set sizes.

Findings

Established relationships between the two types of constants.

Derived explicit values for elementary p-groups.

Linked constants to linear codes and cap set parameters.

Abstract

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group $( G , + )$, a set of weights $W \subset \mathbb{Z}$, and an integral parameter $m$, the $m$-wise Davenport constant with weights $W$ is the smallest integer $n$ such that each sequence over $G$ of length $n$ has at least $m$ disjoint zero-subsums with weights $W$. And, for an integral parameter $d$, the $d$-constrained Davenport constant with weights $W$ is the smallest $n$ such that each sequence over $G$ of length $n$ has a zero-subsum with weights $W$ of size at most $d$. First, we establish a link between these two types of constants and several basic and general results on them. Then, for elementary $p$-groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants.