Optimal $(r,\delta)$-LRCs from monomial-Cartesian codes and their subfield-subcodes

TL;DR

This paper constructs new optimal locally recoverable codes (LRCs) using monomial-Cartesian codes and their subfield-subcodes, providing a method to generate infinitely many such codes with optimal parameters.

Contribution

It introduces a novel approach to constructing $(r, ext{delta})$-optimal LRCs from monomial-Cartesian codes and their subfield-subcodes, expanding the known set of optimal codes.

Findings

Identified conditions for $(r, ext{delta})$-optimal LRCs from MCCs.

Constructed infinitely many new $(r, ext{delta})$-optimal LRCs.

Demonstrated that subfield-subcodes preserve optimal parameters over smaller fields.

Abstract

We study monomial-Cartesian codes (MCCs) which can be regarded as $(r,\delta)$-locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to $(r,\delta)$-optimal LRCs for that distance, which are in fact $(r,\delta)$-optimal. A large subfamily of MCCs admits subfield-subcodes with the same parameters of certain optimal MCCs but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $(r,\delta)$-optimal LRCs and their parameters.