List Decoding of Locally Repairable Codes

TL;DR

This paper introduces an efficient list decoding method for locally repairable codes (LRCs), surpassing traditional bounds by leveraging local error correction, with practical algorithms and analysis for Reed-Solomon based LRCs.

Contribution

It presents a novel list decoding algorithm for LRCs that exceeds the Johnson radius, including explicit constructions and probabilistic decoding analysis.

Findings

Decoding radius surpasses Johnson bound for LRCs

Efficient algorithms for Reed-Solomon based LRCs

Probabilistic decoding achieves high success probability

Abstract

We show that locally repairable codes (LRCs) can be list decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error correction capabilities. The new decoding radius is derived and the asymptotic behavior is analyzed. We give a general list decoding algorithm for LRCs that achieves this radius along with an explicit realization for a class of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic algorithm for unique decoding of low complexity is given and its success probability analyzed.