$S_h$-sets and linear codes over $\mathbb{F}_q$

TL;DR

This paper introduces the concept of $S_h$-linear sets in finite vector spaces, establishing a link with linear codes to derive bounds on the size of such sets.

Contribution

It generalizes $S_h$-sets to vector spaces over finite fields and connects them with linear codes to obtain new bounds.

Findings

Established a correspondence between $q$-ary linear codes and $S_h$-linear sets.

Derived lower bounds for the maximum size of $S_h$-sets in finite vector spaces.

Extended the concept of $S_h$-sets from Abelian groups to finite vector spaces.

Abstract

Let $(G,+)$ be an Abelian group. Given $h\in \mathbb{Z}^+$, a non-empty subset $A$ of $G$ is called an $S_h$-set if all the sums of $h$ distinct elements of $A$ are different. We extend the concept of $S_h$-set to a more general context in the context of finite vectorial spaces over finite fields. More precisely, a $\emptyset \neq A\subseteq \mathbb{F}_q^r$ is called an $S_h$-linear set if all the linear combinations of $h$ elements of $A$ are different. We establish a correspondence between $q$-ary linear codes and $S_h$-linear sets. This connection allow us to find lower bounds for the maximum size of $S_h$-sets in $\mathbb{F}_q^r$.