  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.
