On Error Decoding of Locally Repairable and Partial MDS Codes

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 new 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 analyzed. The second part of this work considers error decoding of LRCs and partial MDS (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 with the probability of successful decoding approaching 1, when the code length goes to infinity.

Decoding of Interleaved Alternant Codes

“…for the interleaved decoding algorithm of [8], [37] when applied to interleaved alternant codes for uniformly distributed errors of a given weight. In the following we present and discuss some numerical results, where we compare these upper and lower bounds 3 . In order to better emphasize the individual contributions of failures and miscorrections, we further include an upper bound on the probability of miscorrection P misc , given in the Appendix, in the plots of Figs.…”

“…However, for a variety of algebraic interleaved codes, it is possible to correct a larger fraction of errors by adopting a collaborative approach. For this reason, interleaved codes have many applications in which burst errors occur naturally or artificially, for instance replicated file disagreement location [1], correcting burst errors in data-storage applications [2], [3], outer codes in concatenated codes [1], [4]- [8], ALOHA-like random-access schemes [5], decoding non-interleaved codes beyond half-the-minimum distance by power decoding [9]- [12], and code-based cryptography [13], [14].…”

Partial MDS Codes with Local Regeneration

“…Recently several constructions of LRCs which maximize the distance have been proposed. However, when considering the mean time to data loss as the performance metric, distance-optimal LRCs are not necessarily optimal, as it is possible to tolerate many failure patterns involving a larger number of nodes than the number that can be guaranteed, while still fulfilling the locality constraints [11], [12]. Partial MDS (PMDS) codes [13]- [15], also referred to as maximally recoverable codes [16], [17], are a subclass of LRCs which guarantee to tolerate all failure patterns possible under these constraints and thereby maximize the mean time to data loss.…”

Partial MDS Codes with Regeneration

“…Recently several constructions of LRCs which maximize the distance have been proposed. However, when considering the mean time to data loss as the performance metric, distance-optimal LRCs are not necessarily optimal, as it is possible to tolerate many failure patterns involving a larger number of nodes than the number that can be guaranteed, while still fulfilling the locality constraints [12], [13]. Partial MDS (PMDS) codes [14], [15], [16], also referred to as maximally recoverable codes [17], [18], are a subclass of LRCs which guarantee to tolerate all failure patterns possible under these constraints and thereby maximize the mean time to data loss.…”