In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $\mathbb{F}_{2^r}$. These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes. In some cases, they are easier to define than their Hermitian counterparts and consequently have a better asymptotic bound on the code rate.

翻译：本文引入特征任意域上的曲线提升码，其灵感来源于$\mathbb{F}_{2^r}$上的Hermitian提升码。这类码专为局部性与可用性设计，其具体参数取决于曲线的选择及其性质。由于构造方式，曲线与直线交点的有理点数量起到关键作用。为阐明这一点并构建具有高可用性的局部恢复码（LRCs）新族，我们重点研究了范数-迹提升码。在某些情形下，相较于Hermitian提升码，范数-迹提升码更易定义，因而在码率渐近界上具有更优表现。