Error Decoding of Locally Repairable and Partial MDS Codes

TL;DR

This paper introduces efficient list-decoding methods for locally repairable and partial MDS codes, leveraging local error correction and interleaved decoding to improve decoding radius and success probability.

Contribution

It presents a general list-decoding algorithm for LRCs, including explicit realizations, and analyzes probabilistic and interleaved decoding strategies for LRCs and PMDS codes.

Findings

List-decoding radius exceeds Johnson radius for LRCs.

Probabilistic unique decoding success probability approaches 1 asymptotically.

Interleaved decoding extends decoding radius for PMDS codes beyond minimum distance.

Abstract

In this work it is shown 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 corresponding decoding radius is derived and the asymptotic behavior is analyzed. A general list-decoding algorithm for LRCs that achieves this radius is proposed along with an explicit realization for LRCs that are subcodes of Reed--Solomon codes (such as, e.g., Tamo--Barg LRCs). Further, a probabilistic algorithm of low complexity for unique decoding of LRCs is given and its success probability is analyzed. The second part of this work considers error decoding of LRCs and partial maximum distance separable (PMDS) codes through interleaved decoding. For a specific class of LRCs the success probability of interleaved decoding is investigated. For PMDS codes, it is shown that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance such that the probability of successful decoding approaches $1$ when the code length goes to infinity.