Optimal (r,δ)-LRCs from monomial-Cartesian codes and their subfield-subcodes

• Published in: Designs, codes, and cryptography Vol. 92; no. 9; pp. 2549 - 2586
• Main Authors: Galindo, C., Hernando, F., Martín-Cruz, H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 2024; Springer Nature B.V
• Subjects: Cartesian coordinates; Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Parameters; Secondary 94B27; Monomial-Cartesian codes; Locally recoverable codes; Subfield-subcodes; Primary 94B05; 14G50
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-024-01403-z

We study monomial-Cartesian codes (MCCs) which can be regarded as
(
r
,
δ
)
-locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to
(
r
,
δ
)
-optimal LRCs for that distance, which are in fact
(
r
,
δ
)
-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
,
δ
)
-optimal LRCs and their parameters.