Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

TL;DR

This paper establishes new upper bounds on the dimension of binary linear locally repairable codes and provides constructions that meet these bounds, improving understanding of code limits and design.

Contribution

It introduces two novel upper bounds on the dimension of binary LRCs, including an explicit bound for certain parameters, and offers constructions achieving the second bound.

Findings

The first bound generalizes previous bounds for disjoint repair groups.

The second bound outperforms the Cadambe-Mazumdar bound for certain parameters.

Explicit constructions attain the second bound for codes with minimum distance at least 6.

Abstract

For binary $[n,k,d]$ linear locally repairable codes (LRCs), two new upper bounds on $k$ are derived. The first one applies to LRCs with disjoint local repair groups, for general values of $n,d$ and locality $r$, containing some previously known bounds as special cases. The second one is based on solving an optimization problem and applies to LRCs with arbitrary structure of local repair groups. Particularly, an explicit bound is derived from the second bound when $d\geq 5$. A specific comparison shows this explicit bound outperforms the Cadambe-Mazumdar bound for $5\leq d\leq 8$ and large values of $n$. Moreover, a construction of binary linear LRCs with $d\geq6$ attaining our second bound is provided.