Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Viviana Carolina Guerrero Pantoja, John H. Castillo, Carlos Alberto Trujillo Solarte

TL;DR

The paper addresses the problem of bounding the maximal size of $S_h$-sets in finite vector spaces by linking them to $q$-linear codes with minimum distance $d\ge 2h+1$. It introduces $S_h$-linear sets in $\mathbb{F}_q^r$ and proves a one-to-one correspondence with such codes via the parity-check matrix, yielding translates between additive combinatorics and coding theory parameters. This correspondence leads to practical lower bounds for $S_h$-sets and, equivalently, lower bounds on code dimensions given a fixed length and distance, computed through minimal redundancy $r=n-k$ and Grassl codetables, as well as constructions from BCH codes. The results extend known binary connections to general $q$ and provide a framework for deriving bounds and guiding future open problems in both domains.

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 $\mathbb{F}_q^r$.

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

TL;DR

The paper addresses the problem of bounding the maximal size of -sets in finite vector spaces by linking them to -linear codes with minimum distance . It introduces -linear sets in and proves a one-to-one correspondence with such codes via the parity-check matrix, yielding translates between additive combinatorics and coding theory parameters. This correspondence leads to practical lower bounds for -sets and, equivalently, lower bounds on code dimensions given a fixed length and distance, computed through minimal redundancy and Grassl codetables, as well as constructions from BCH codes. The results extend known binary connections to general and provide a framework for deriving bounds and guiding future open problems in both domains.

Abstract

Let be an Abelian group. Given , a non-empty subset of is called an -set if all the sums of distinct elements of are different. We extend the concept of -set to a more general context in the context of finite vectorial spaces over finite fields. More precisely, a is called an -linear set if all the linear combinations of elements of are different. We establish a correspondence between -ary linear codes and -linear sets. This connection allow us to find lower bounds for the maximum size of -sets in .

Paper Structure

This paper contains 4 sections, 15 theorems, 22 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A correspondence between $q$-linear codes and $S_h$-sets
  4. Consequences of Theorem \ref{['principalteor']}

Key Result

Theorem 2.1

If $H$ is a parity-check matrix of a code $\mathcal{C}$ with length $n$, then $\mathcal{C}$ has minimum distance $d$ if and only if any set with $d-1$ columns of $H$ is a linearly independent set, and there exists a set with $d$ columns of $H$ that is linearly dependent.

Figures (2)

  • Figure 1: Values of $\mathcal{V}_2(h,n)$ for $2\leq h\leq 6$ and $2h+1\leq n\leq 256$.
  • Figure 2: Values of $\overline{\mathcal{V}}_3(h,n)$ for $2\leq h\leq 6$ and $2h+1\leq n\leq 243$.

Theorems & Definitions (32)

  • Theorem 2.1: LX
  • Theorem 2.2: Singleton bound
  • Definition 3.1: $h$-linear combination
  • Definition 3.2: $S_{h}$-linear set
  • Example 3.1
  • Lemma 3.1
  • proof
  • Example 3.2
  • Proposition 3.1
  • proof
  • ...and 22 more