Locally recoverable $J$-affine variety codes
Carlos Galindo, Fernando Hernando, Carlos Munuera
TL;DR
The paper addresses the repair problem for distributed storage by designing long locally recoverable codes with robust multi-erasure recovery. It constructs LRC codes as subfield-subcodes of $J$-affine variety codes, using evaluation on zero sets $Z_J$ and trace maps to obtain codes over $bF_q$, with dimensions controlled by carefully chosen index sets $ abla$. The main contributions include explicit locality bounds and optimality results via cyclotomic orbit techniques, plus detailed univariate and bivariate constructions that yield codes with lengths $n obreak o o obreak q$ in many cases and often $(r,δ)$-optimality; several examples achieve sharpness and strong minimum-distance behavior. The work provides practical, parameter-rich schemes for long LRC codes suitable for scalable distributed storage, expanding the toolkit beyond small-length codes and offering concrete recovery-set structures based on algebraic-geometric constructions.
Abstract
A locally recoverable (LRC) code is a code over a finite field $\mathbb{F}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some $J$-affine variety codes. For these LRC codes, we compute localities $(r, δ)$ that determine the minimum size of a set $\bar{R}$ of positions so that any $δ- 1$ erasures in $\bar{R}$ can be recovered from the remaining $r$ coordinates in this set. We also show that some of these LRC codes with lengths $n\gg q$ are $(δ-1)$-optimal.