Lower Rate Bounds for Hermitian-Lifted Codes for Odd Prime Characteristic

TL;DR

This paper establishes a positive lower bound on the rate of Hermitian-lifted locally recoverable codes with high availability over fields of odd prime characteristic, generalizing previous results from characteristic 2.

Contribution

It extends the rate lower bound analysis of Hermitian-lifted codes from characteristic 2 to any odd prime characteristic, providing a broader understanding of their efficiency.

Findings

Rate of Hermitian-lifted codes is bounded below by a positive constant depending on p.

Generalization from characteristic 2 to odd prime characteristics.

Supports the design of efficient locally recoverable codes in diverse field settings.

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 Lopez, Malmskog, Matthews, Pi\~nero-Gonzales, 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. Lopez et al. lower bounded 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 $q=p^l$ for any odd prime $p$.