Constructions of Optimal and Almost Optimal Locally Repairable Codes

TL;DR

This paper constructs almost optimal and optimal locally repairable codes (LRCs) with specific parameters, addressing open problems in the field and expanding the types of codes available over small finite fields.

Contribution

It introduces new constructions of almost optimal linear LRCs for cases where (r+1) f n, including some that are proven to be optimal, over small finite fields.

Findings

Constructed almost optimal linear LRCs with minimum distance within one of the optimal bound.

Developed optimal LRCs that do not require large finite fields for certain parameters.

Addressed open problems in LRC construction for cases where f(r+1) mid n.

Abstract

Constructions of optimal locally repairable codes (LRCs) in the case of $(r+1) \nmid n$ and over small finite fields were stated as open problems for LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and connections to matroid theory", \emph{2013 IEEE ISIT}]. In this paper, these problems are studied by constructing almost optimal linear LRCs, which are proven to be optimal for certain parameters, including cases for which $(r+1) \nmid n$. More precisely, linear codes for given length, dimension, and all-symbol locality are constructed with almost optimal minimum distance. `Almost optimal' refers to the fact that their minimum distance differs by at most one from the optimal value given by a known bound for LRCs. In addition to these linear LRCs, optimal LRCs which do not require a large field are constructed for certain classes of parameters.