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

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.

arXiv commentThis is a revised version of the manuscript "Optimal $(r,\delta )$-LRCs from zero-dimensional affine variety codes and their subfield-subcodes". We have modified the title and the new one is "Optimal $(r,\delta )$-LRCs from monomial-Cartesian codes and their subfield-subcodes". This new version contains rather changes, the main ones appear in Section 4

References in corpus (11)

• A family of optimal locally recoverable codes
• Binary Cyclic Codes that are Locally Repairable
• Affine cartesian codes
• Locally recoverable codes from algebraic curves and surfaces
• Constructions of Locally Recoverable Codes which are Optimal
• Stabilizer quantum codes from $J$-affine variety codes and a new Steane-like enlargement
• Quantum codes from affine variety codes and their subfield-subcodes
• On the distance of stabilizer quantum codes from $J$-affine variety codes
• Polar decreasing monomial-Cartesian codes
• Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes
• Locally recoverable $J$-affine variety codes

Cited by in corpus (3)

• Quantum $(r,\delta )$-locally recoverable codes
• MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes
• Impure codes exceeding the pure bounds for quantum local recovery