$S_h$-sets and linear codes over $\mathbb{F}_q$
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| Format: | Preprint |
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2024
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| _version_ | 1866914086319554560 |
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| author | Pantoja, Viviana Carolina Guerrero Castillo, John H. Solarte, Carlos Alberto Trujillo |
| author_facet | Pantoja, Viviana Carolina Guerrero Castillo, John H. Solarte, Carlos Alberto Trujillo |
| contents | 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 $\mathbb{F}_q^r$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $S_h$-sets and linear codes over $\mathbb{F}_q$ Pantoja, Viviana Carolina Guerrero Castillo, John H. Solarte, Carlos Alberto Trujillo Number Theory 11B13, 94B05, 94B65 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 $\mathbb{F}_q^r$. |
| title | $S_h$-sets and linear codes over $\mathbb{F}_q$ |
| topic | Number Theory 11B13, 94B05, 94B65 |
| url | https://arxiv.org/abs/2411.19413 |