A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

TL;DR

This paper establishes a new lower bound on the field size for locally repairable codes, constructs cyclic maximally recoverable codes meeting this bound, and identifies conditions for cyclic permutation of known MR codes.

Contribution

It introduces a new bound on field size for LRCs, constructs cyclic MR codes achieving this bound, and characterizes when known MR codes can be made cyclic.

Findings

New lower bound on field size for LRCs

Construction of cyclic MR codes with optimal field size

Conditions for permuting non-cyclic MR codes to cyclic form

Abstract

We prove a new lower bound on the field size of locally repairable codes (LRCs). Additionally, we construct maximally recoverable (MR) codes which are cyclic. While a known construction for MR codes has the same parameters, it produces non-cyclic codes. Furthermore, we prove both necessary conditions and sufficient conditions that specify when the known non-cyclic MR codes may be permuted to become cyclic, thus proving our construction produces cyclic MR codes with new parameters. Furthermore, using our new bound on the field size, we show that the new cyclic MR codes have optimal field size in certain cases. Other known LRCs are also shown to have optimal field size in certain cases.