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Lower rate bounds for Hermitian-lifted codes in odd prime characteristic

Beth Malmskog and Na’ama Nevo

Vol. 18 (2025), No. 4, 643–664
Abstract

Locally recoverable codes are error-correcting codes with the additional property that every symbol of any codeword can be recovered from a small set of other symbols. This property is particularly desirable in cloud storage applications. A locally recoverable code is said to have availability t if each position has t disjoint recovery sets. Hermitian-lifted codes are locally recoverable codes with high availability first described by López, Malmskog, Matthews, Piñero-Gonzáles, and Wootters. The codes are based on the well-known Hermitian curve and incorporate the novel technique of lifting to increase the rate of the code. López et al. (2021) provided a lower bound for the rate of the codes defined over fields with characteristic 2. This paper generalizes their work to show that the rate of Hermitian-lifted codes is bounded below by a positive constant depending on p, when q = pl for any odd prime p.

Keywords
Hermitian-lifted codes, locally recoverable codes, availability, Hermitian curve
Mathematical Subject Classification
Primary: 11T71, 14G50, 94B27
Milestones
Received: 28 August 2023
Revised: 29 January 2024
Accepted: 6 March 2024
Published: 30 July 2025

Communicated by Nathan Kaplan
Authors
Beth Malmskog
Mathematics and Computer Science
Colorado College
Colorado Springs, CO
United States
Na’ama Nevo
Mathematics
Northeastern University
Boston, MA
United States