Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Repairing Reed-Solomon Codes with Side Information

Thi Xinh Dinh, Ba Thong Le, Son Hoang Dau, Serdar Boztas, Stanislav Kruglik, Han Mao Kiah, Emanuele Viterbo, Tuvi Etzion, Yeow Meng Chee

TL;DR

Repairs of a single erased Reed-Solomon symbol are generalized to exploit side information modeled as an $\mathbb{F}_q$-subspace $\mathcal{S}$ of dimension $s$; the authors develop a trace-based repair framework and prove that the minimum repair bandwidth depends only on $s$ (not on the specific content of $S$), deriving a general lower bound and constructing optimal subspace-polynomial schemes in several parameter regimes. In the full-length setting with $n=q^{\ell}$ and $n-k=q^{m}$, the lower bound specializes to an explicit bandwidth $(q^{\ell}-1)(\ell-s)-\dfrac{(q^{\ell-s}-1)(q^{m}-1)}{q-1}$, and suitable choices of subspaces yield optimal schemes. The work reduces the repair problem with side information to a subspace-intersection optimization, linking bandwidth performance to geometric properties of subspace intersections. These results enable lower-cost repairs in distributed storage when side information is available and provide a concrete framework for designing bandwidth-efficient RS repairs under side information.

Abstract

We generalize the problem of recovering a lost/erased symbol in a Reed-Solomon code to the scenario in which some side information about the lost symbol is known. The side information is represented as a set $S$ of linearly independent combinations of the sub-symbols of the lost symbol. When $S = \varnothing$, this reduces to the standard problem of repairing a single codeword symbol. When $S$ is a set of sub-symbols of the erased one, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on $|S|$ and not the content of $S$ and construct a lower bound on the repair bandwidth of a linear repair scheme with side information $S$. We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. Finally, we demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes.

Repairing Reed-Solomon Codes with Side Information

TL;DR

Repairs of a single erased Reed-Solomon symbol are generalized to exploit side information modeled as an -subspace of dimension ; the authors develop a trace-based repair framework and prove that the minimum repair bandwidth depends only on (not on the specific content of ), deriving a general lower bound and constructing optimal subspace-polynomial schemes in several parameter regimes. In the full-length setting with and , the lower bound specializes to an explicit bandwidth , and suitable choices of subspaces yield optimal schemes. The work reduces the repair problem with side information to a subspace-intersection optimization, linking bandwidth performance to geometric properties of subspace intersections. These results enable lower-cost repairs in distributed storage when side information is available and provide a concrete framework for designing bandwidth-efficient RS repairs under side information.

Abstract

We generalize the problem of recovering a lost/erased symbol in a Reed-Solomon code to the scenario in which some side information about the lost symbol is known. The side information is represented as a set of linearly independent combinations of the sub-symbols of the lost symbol. When , this reduces to the standard problem of repairing a single codeword symbol. When is a set of sub-symbols of the erased one, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on and not the content of and construct a lower bound on the repair bandwidth of a linear repair scheme with side information . We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. Finally, we demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes.
Paper Structure (15 sections, 16 theorems, 26 equations, 1 figure, 1 algorithm)

This paper contains 15 sections, 16 theorems, 26 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Notations
  4. Trace Repair Method
  5. Recovering an Erased Symbol with Side Information
  6. The Problem Description
  7. Optimal Repair Bandwidths Only Depend on the Side Information Set Size
  8. A Lower Bound on the Bandwidth with Side Information
  9. Optimal Subspace-Polynomial-Based Repair Schemes
  10. Bandwidth Reductions Given Side Information
  11. Conclusions
  12. Appendix
  13. Proof of Proposition \ref{['pro:scheme_corresponds_polys']}
  14. Proof of Proposition \ref{['prop:lower_bound']}
  15. A discussion on the subspace intersections with the lowest repair bandwidth

Key Result

Proposition 1

Let $S \stackrel{\hbox{\tiny $\triangle$}}{=} \{\bm{\beta}_i\}_{i\in [s]}$ be a linearly independent set and $f(\bm{\alpha}^*)$ be a symbol of Reed-Solomon code RS$(n,k)$ over $\mathbb{F}_{q^{\ell}}$, $n \leq q^\ell$. A linear repair scheme for $f(\bm{\alpha}^*)$ with side information $S$ correspond

Figures (1)

  • Figure 1: An illustration of repair schemes that recover $\bm{a}\space =\space (a_1, a_2)$ with and without side information. The side information $a_1\space+\space a_2$ leads to a reduction of 1 bit in the repair bandwidth. The repair node first obtains $a_2\space\leftarrow\space (a_2\space+\space b_1)\space-\space b_1$, and then $a_1 \leftarrow (a_1+a_2)-a_2$.

Theorems & Definitions (35)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Corollary 1
  • proof
  • Lemma 2
  • ...and 25 more