Explicit optimal-length locally repairable codes of distance 5
Allison Beemer, Ryan Coatney, Venkatesan Guruswami, Hiram H. López, Fernando Piñero
TL;DR
This work addresses the problem of constructing explicit optimal-length locally repairable codes with distance $d=5$ over an alphabet $F_q$, achieving length growing as $O(q^2)$. It uses Cartesian codes on two components evaluating polynomials over the grid $V=F_q^* \times F_q^*$ and selects a subspace by excluding monomials with $i \equiv r (mod r+1)$ to enforce locality. The contributions include an explicit code $C(V,L)$ of length $n=(q-1)^2$ with distance at least $5$ and dimension $n - n/(r+1) - 3$, meeting the Singleton-like bound and hence optimal for $r>3$; recovery sets are given by the partition into $\alpha R \times {\gamma}$ with $R = {\beta in F_q^* : {\beta}^{r+1}=1}$. This work provides the first explicit, optimal-length LRCs for distance 5 that scale quadratically in $q$, clarifying the alphabet-size versus block-length trade-off in LRC design and offering practical templates for robust storage systems.
Abstract
Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give explicit constructions of optimal-length (in terms of alphabet size), optimal LRCs with minimum distance equal to 5.