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arXiv · 2109.14478

Quadratic-Curve-Lifted Reed-Solomon Codes

Hedongliang Liu·Lukas Holzbaur·Nikita Polyanskii·Sven Puchinger·Antonia Wachter-Zeh
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Abstract

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.

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Hedongliang Liu, Lukas Holzbaur, Nikita Polyanskii, Sven Puchinger, Antonia Wachter-Zeh. 2021-09-29. Quadratic-Curve-Lifted Reed-Solomon Codes. https://arxiv.org/abs/2109.14478

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