Repairable Fountain Codes

TL;DR

This paper introduces a new family of systematic Fountain codes with sparse parities, enabling efficient data recovery and local repair in distributed storage systems through probabilistic analysis of sparse random matrices.

Contribution

The paper presents a novel construction of Fountain codes with sparse parities and logarithmic locality, along with a mathematical analysis of their rank properties.

Findings

Codes can recover original data from slightly more than k encoded symbols.

Single symbol repair requires accessing only O(log k) other symbols.

High probability of perfect matchings in associated sparse bipartite graphs.

Abstract

We introduce a new family of Fountain codes that are systematic and also have sparse parities. Given an input of $k$ symbols, our codes produce an unbounded number of output symbols, generating each parity independently by linearly combining a logarithmic number of randomly selected input symbols. The construction guarantees that for any $\epsilon>0$ accessing a random subset of $(1+\epsilon)k$ encoded symbols, asymptotically suffices to recover the $k$ input symbols with high probability. Our codes have the additional benefit of logarithmic locality: a single lost symbol can be repaired by accessing a subset of $O(\log k)$ of the remaining encoded symbols. This is a desired property for distributed storage systems where symbols are spread over a network of storage nodes. Beyond recovery upon loss, local reconstruction provides an efficient alternative for reading symbols that cannot be accessed directly. In our code, a logarithmic number of disjoint local groups is associated with each systematic symbol, allowing multiple parallel reads. Our main mathematical contribution involves analyzing the rank of sparse random matrices with specific structure over finite fields. We rely on establishing that a new family of sparse random bipartite graphs have perfect matchings with high probability.