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 $p$.

翻译：局部可恢复码是一种纠错码，具有附加性质：任意码字的每个符号均可通过少量其他符号恢复。该性质在云存储应用中尤为重要。若每个位置都存在t个互不相交的恢复集，则称该局部可恢复码具有可用性t。厄米特提升码是Lopez、Malmskog、Matthews、Piñero-Gonzales与Wootters首次描述的一类高可用性局部可恢复码。该类码基于著名的厄米特曲线，并结合了提升这一新颖技术以提高码率。Lopez等人给出了特征为2的域上定义码的速率下界。本文将他们的工作推广至奇素数p的情形，证明当q=p^l时，厄米特提升码的速率被一个仅依赖于p的正常数所界定。