Mathematical LoRE: Local Recovery of Erasures using Polynomials, Curves, Surfaces, and Liftings

TL;DR

The paper surveys how underlying algebraic-geometric structures can be leveraged to construct locally recoverable codes (LRCs) with small repair sets and multiple disjoint recovery options. It surveys polynomial LRCs via good polynomials and the TB construction, curve-based LRCs via the TBV framework and Hermitian-lifted codes, and availability-enhanced constructions using fiber products (including GK curves and Artin-Schreier families) and lifted codes. It also discusses bounds and optimal constructions, showing that LRCs from curves and surfaces can exceed traditional MDS-like limits, with curve-based codes potentially achieving long lengths over small fields and surface-based codes illustrating the breakdown of simple length conjectures. The work highlights the practical relevance for distributed storage and the rich interplay between algebraic geometry and coding theory, offering a toolkit for designing high-availability, high-rate LRCs. Key results include explicit polynomial LRCs with optimal locality, TBV curve-based LRCs with tunable locality and availability, GK and Artin-Schreier constructions with multiple recovery sets, and Hermitian-lifted codes achieving high rates and availability on Hermitian curves, together with bounds showing where optimal LRCs can be longer than classical MDS codes.

Abstract

Employing underlying geometric and algebraic structures allows for constructing bespoke codes for local recovery of erasures. We survey techniques for enriching classical codes with additional machinery, such as using lines or curves in projective space for local recovery sets or products of curves to enhance the availability of data.

Figures (5)

• Figure 1: Coordinates of a received word from polynomial-based LRC grouped according to the partition of $\mathbb{F}_{13}^*$ given by the polynomial $g(x)=x^3$.
• Figure 2: Generalized GK curve over $\mathbb{F}_{729}$ as a fiber product, which gives rise to LRCs with availability $2$.
• Figure 3: On the generalized GK curve $X_3$ over $\mathbb{F}_{729}$, two disjoint recovery sets for an erasure at the point $(P,Q)$ naturally arise from the fiber product structure, one comprised of points $(P,Q')$, where $Q' \neq Q$, and other other of points $(P',Q)$, where $P' \neq P$ as specified in Example \ref{['E:GK_ex']}.
• Figure 4: Exponent pairs $(a,b)$ with $x^ay^b$ for $q=8$ ($a$ is on horizontal axis) lifted
• Figure 5: Exponents of some good monomials $x^ay^b$ for Hermitian-lifted and norm-trace-lifted codes over $\mathbb{F}_{64}$ defined using $\mathcal{H}_8$ and $\mathcal{X}_{2,6}$nt_lifted_binary

Theorems & Definitions (15)

• Definition 1
• Theorem 2
• Definition 3
• Example 4
• Example 5
• Definition 6
• Example 7
• Example 8
• Definition 9
• Example 10