Lifted codes are a class of evaluation codes attracting more attention due to good locality and intermediate availability. In this work we introduce and study quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, where the codeword symbols whose coordinates are on a quadratic curve form a codeword of a Reed-Solomon code. We first develop a necessary and sufficient condition on the monomials which form a basis the code. Based on the condition, we give upper and lower bounds on the dimension and show that the asymptotic rate of a QC-LRS code over $\mathbb{F}_q$ with local redundancy $r$ is $1-\Theta(q/r)^{-0.2284}$. Moreover, we provide analytical results on the minimum distance of this class of codes and compare QC-LRS codes with lifted Reed-Solomon codes by simulations in terms of the local recovery capability against erasures. For short lengths, QC-LRS codes have better performance in local recovery for erasures than LRS codes of the same dimension.

翻译：取消的代码是一类评价代码,由于地点和可用性强而引起更多关注。在这项工作中,我们引入并研究二次曲线-曲线提升的Reed-Solomon(QC-LRS)代码,其坐标在二次曲线上的代码符号形成Reed-Solomon代码的代码。我们首先对构成代码基础的单项代码开发一个必要和充分的条件。根据条件,我们给出了尺寸上下限,并表明,对于本地冗余值,QC-LRS代码的无时效率为$\mathbb{F ⁇ q$$1\Theta(q/r) ⁇ _0.2284}$$。此外,我们提供了这一类代码最低距离的分析结果,并将QC-LRS代码与取消的Reed-Solomon代码相比较。根据条件,对本地的恢复能力进行模拟,以弥补消蚀值为标准,从短期来看,QC-LRS代码在本地的消化过程中的恢复效果优于同一尺寸的LRS码。