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Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

Kathryn Haymaker, Beth Malmskog, Gretchen L. Matthews

TL;DR

This work addresses constructing locally recoverable codes with both hierarchical locality and availability by exploiting underlying geometry in Reed-Muller codes and fiber-product codes. It shows that nested affine subspaces in RM codes and the fiber-product structure of curves yield natural two-level and multi-level hierarchies, and that fiber-product codes can be viewed as punctured subcodes of Reed-Muller codes, unifying the constructions. The authors derive explicit hierarchical parameters for RM-based and fiber-product-based LRCs, including two notable specializations via Artin-Schreier and Hermitian curves, and extend the framework to H-level hierarchies with availability at each level. The results highlight the versatility and flexibility of geometry-driven recovery strategies, providing a foundation for more refined hierarchy-availability bounds and broader applicability to algebraic-geometry code families.

Abstract

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed-Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

TL;DR

This work addresses constructing locally recoverable codes with both hierarchical locality and availability by exploiting underlying geometry in Reed-Muller codes and fiber-product codes. It shows that nested affine subspaces in RM codes and the fiber-product structure of curves yield natural two-level and multi-level hierarchies, and that fiber-product codes can be viewed as punctured subcodes of Reed-Muller codes, unifying the constructions. The authors derive explicit hierarchical parameters for RM-based and fiber-product-based LRCs, including two notable specializations via Artin-Schreier and Hermitian curves, and extend the framework to H-level hierarchies with availability at each level. The results highlight the versatility and flexibility of geometry-driven recovery strategies, providing a foundation for more refined hierarchy-availability bounds and broader applicability to algebraic-geometry code families.

Abstract

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed-Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.
Paper Structure (17 sections, 10 theorems, 34 equations, 3 figures)

This paper contains 17 sections, 10 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Locality, availability, and code constructions
  3. Background and notation
  4. Reed-Muller and fiber product codes
  5. Relating fiber product codes and Reed-Muller codes
  6. Two-level hierarchical recovery of Reed-Muller and generalized fiber product constructions
  7. Hierarchical recovery of Reed-Muller codes
  8. Hierarchical recovery of fiber product codes
  9. Hierarchical recovery of LRC($t$) from a fiber product of Artin-Schreier curves
  10. Hierarchical recovery of LRC($2$) from the Hermitian curve as a fiber product
  11. Codes with $H$-level hierarchy from Reed-Muller and generalized fiber product construction
  12. $H$-level hierarchy from Reed-Muller codes
  13. $H$-level hierarchy from fiber product codes
  14. Combined hierarchy and availability
  15. Reed-Muller codes with hierarchy and availability
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let curves $\{\mathcal{Y}_j\}_{j\in\left\{ 1, \dots, t \right\}}$, $\mathcal{Y}$, maps $\{h_j:\mathcal{Y}_j \rightarrow\mathcal{Y}\}_{j\in\left\{ 1, \dots, t \right\}}$, a divisor $D$ on $\mathcal{Y}(\mathbb{F}_q)$, and sets $S\subseteq\mathcal{Y}(\mathbb{F}_q)$ and $B=g^{-1}(S)$ be all as described

Figures (3)

  • Figure 1: This is a cartoon illustrating a code with 2-level hierarchy. Small repair groups of size 6 can recover a single erasure, while larger repair groups of size 12 can recover up to 4 erasures.
  • Figure 2: This cartoon illustrates how some hierarchical repair groups naturally arise in the evaluation points of $RM_5(v,3)$ for $v\leq 3$. The positions corresponding to points of any $\mathbb{F}_5$-rational line passing through the given point can act as a repair group for local recovery, with a middle code arising from any plane containing the line. Here we see one way to partition the space into planes, the points of which correspond to the positions in the support of middle codes. For each point in these planes, there are six lines in the plane passing through the point (four are illustrated here), any of which can be chosen to give the positions in the support of the lower code for the chosen position.
  • Figure 3: A diagram of covers in the iterated fiber product construction. The tower of covers leading to $t$-level hierarchy is the far left sequence of covers.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2.1: haymaker2018locally
  • Definition 5: sasidharan2015codes
  • Theorem 3.1
  • proof
  • Example 1: Two-level hierarchical recovery using a Reed-Muller code
  • Remark 1
  • ...and 22 more