Repairing Reed-Solomon Codes with Side Information

TL;DR

This paper extends Reed-Solomon code repair to scenarios with side information, establishing bandwidth bounds and optimizing repair schemes for various parameters, improving efficiency in data recovery.

Contribution

It introduces a generalized repair framework with side information, derives lower bounds on repair bandwidth, and optimizes existing schemes for better performance.

Findings

Repair bandwidth depends on the size of side information set |S|.

Optimal repair bandwidths are achievable in several parameter regimes.

Subspace-polynomial schemes can be optimized by selecting appropriate subspaces.

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.