Good locally repairable codes via propagation rules

TL;DR

This paper introduces new propagation rules for constructing locally repairable codes, leading to optimal codes that surpass existing bounds and simplify previous constructions.

Contribution

It develops novel propagation rules specifically for locally repairable codes, enabling the construction of dimension-optimal and Singleton-optimal codes, and extends classical bounds.

Findings

Concatenation yields dimension-optimal binary locally repairable codes.

Explicit construction of codes exceeding Zyablov-type bound.

Lengthening rules produce optimal codes from Hamming and MDS codes.

Abstract

In classical coding theory, it is common to construct new codes via propagation rules. There are various propagation rules to construct classical block codes. However, propagation rules have not been extensively explored for constructions of locally repairable codes. In this paper, we introduce a few propagation rules to construct good locally repairable codes. To our surprise, these simple propagation rules produce a few interesting results. Firstly, by concatenating a locally repairable code as an inner code with a classical block code as an outer code, we obtain quite a few dimension-optimal binary locally repairable codes. Secondly, from this concatenation, we explicitly build a family of locally repairable codes that exceeds the Zyablov-type bound. Thirdly, by a lengthening propagation rule that adds some rows and columns from a parity-check matrix of a given linear code, we are able to produce a family of dimension-optimal binary locally repairable codes from the extended Hamming codes, and to convert a classical maximum distance separable (MDS) code into a Singleton-optimal locally repairable code. Furthermore, via the lengthening propagation rule, we greatly simplify the construction of a family of locally repairable codes in \cite[Theorem 5]{MX20} that breaks the asymptotic Gilbert-Varshamov bound. In addition, we make use of three other propagation rules to produce more dimension-optimal binary locally repairable codes. Finally, one of phenomena that we observe in this paper is that some trivial propagation rules in classical block codes do not hold anymore for locally repairable codes.