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

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 $\varnothing \ne A\subseteq {\mathbb{F}}_q^r$ 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 ${\mathbb{F}}_q^r$.

17 pages, 2 figures. In this version some mistakes are corrected