Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

TL;DR

This paper develops new constructions of optimal locally repairable codes over small fields, including cyclic and linear codes with various locality and availability properties, enhancing data repair efficiency.

Contribution

It introduces four novel methods for constructing optimal locally repairable codes over small fields, expanding the types and properties of such codes.

Findings

Binary cyclic codes with locality two and availability one.

Binary cyclic codes with multiple repair sets based on Simplex codes.

Non-cyclic optimal binary linear codes with higher locality.

Abstract

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.