Bounds and Constructions of Locally Repairable Codes: Parity-check Matrix Approach

TL;DR

This paper introduces a parity-check matrix framework to analyze bounds and construct optimal locally repairable codes (LRCs), providing new structural insights and classifying all optimal binary LRCs with specific parameters.

Contribution

It presents a unified parity-check matrix approach for analyzing bounds and constructing optimal LRCs, and classifies all possible optimal binary LRCs with certain parameters.

Findings

Derived an upper bound on the minimum distance of optimal LRCs based on field size.

Identified only 5 parameter classes for optimal binary LRCs.

Enumerated all optimal binary LRCs attaining the Singleton-like bound within these classes.

Abstract

A $q$-ary $(n,k,r)$ locally repairable code (LRC) is an $[n,k,d]$ linear code over $\mathbb{F}_q$ such that every code symbol can be recovered by accessing at most $r$ other code symbols. The well-known Singleton-like bound says that $d \le n-k-\lceil k/r\rceil +2$ and an LRC is said to be optimal if it attains this bound. In this paper, we study the bounds and constructions of LRCs from the view of parity-check matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is proposed. Several useful structural properties on $q$-ary optimal LRCs are obtained. We derive an upper bound on the minimum distance of $q$-ary optimal $(n,k,r)$-LRCs in terms of the field size $q$. Then, we focus on constructions of optimal LRCs over binary field. It is proved that there are only 5 classes of possible parameters with which optimal binary $(n,k,r)$-LRCs exist. Moreover, by employing the proposed parity-check matrix approach, we completely enumerate all these 5 classes of possible optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes.