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On the construction of large local arcs

Ferdinand Ihringer, Yue Zhou

Abstract

Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a \emph{local arc} which is defined as a collection $\mathcal{S}$ of disjoint point sets $S_{i}$ in $\mathrm{PG}(2,q)$ such that $S_{i} \cup S_{j}$ is an arc for any $S_{i}, S_{j} \in \mathcal{S}$. We focus on the upper and lower bounds on the sizes of maximum $k$-uniform local arcs. For $q=p^m$ with $p$ prime, we construct $k$-uniform local arcs in $\mathrm{PG}(2,q)$ of size $Ω(q^{d})$ where $d$ is between $1.1167$ and $1.25$ depending only on $m$. For $k=4$, this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in $q$--a significant improvement over the previously known $O(q)$ constructions for such LRCs.

On the construction of large local arcs

Abstract

Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a \emph{local arc} which is defined as a collection of disjoint point sets in such that is an arc for any . We focus on the upper and lower bounds on the sizes of maximum -uniform local arcs. For with prime, we construct -uniform local arcs in of size where is between and depending only on . For , this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in --a significant improvement over the previously known constructions for such LRCs.
Paper Structure (7 sections, 12 theorems, 59 equations)

This paper contains 7 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Upper bounds
  3. Constructions
  4. Constructions for $q$ prime
  5. Constructions for $q$ prime power
  6. Small Numbers
  7. Future Work

Key Result

Theorem 1.1

Let $q=p^m$, $p$ a prime. A maximum $k$-uniform local arc $\mathcal{S}$ in $\mathrm{PG}(2, q)$ has size at most $\left(\min\{1, \frac{\sqrt{2}}{\sqrt{k-1}}\} + o(1)\right) q^{1.5}$ and

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2: Expander-Mixing Lemma for Biregular Graphs
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 10 more