List Decoding of Locally Repairable Codes

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.

On Error Decoding of Locally Repairable and Partial MDS Codes

“…In [13], a family of LRCs, popular for the small required field size and for fulfilling the Singleton-like bound on the distance [9], [11], was constructed as subcodes of (generalized) Reed-Solomon (GRS) codes. In addition, partial maximum distance separable (PMDS) codes 1 [7], [15]- [23] fulfill an even stronger notion of locality by requiring that any information-theoretically solvable erasure pattern can be recovered and have been proposed for use in distributed storage systems to further decrease the probability of data loss.…”

“…KR3517/9-1. Parts of this paper have been presented at the 2018 IEEE International Symposium on Information Theory (ISIT) [1] and 2019 IEEE Information Theory Workshop (ITW) [2]. 1 In [7], [15]- [18] these codes are referred to as maximally recoverable codes, the definitions are equivalent.…”

“…Different approaches have been considered to reduce the complexity of repairing such systems in the case of node failures, with special attention to the more likely event of a single or very few node failures, in which case an efficient repair is especially important. The most prominent approaches addressing this issue are locally repairable codes (LRC) [1]- [5], which limit the number of nodes involved in the repair, and regenerating codes [6], [7], which aim to decrease the network traffic required for repair. The main motivation of storage codes such as LRCs and regenerating codes is erasure correction, as these occur naturally in distributed storage systems whenever nodes fail, e.g., due to hardware failures, power outages or maintenance.…”

“…Efficient repair of nodes by error correction from parts of the received word was considered in [7]. In [5] it was shown that the error correction radius of LRCs can be increased beyond the Johnson radius.…”

Locality in Crisscross Error Correction

“…It was shown in [11] that codes in the cover metric can be list decoded up to a Johnson-like upper bound. In [16] it was shown that for some parameters block LRCs can be list decoded beyond the Johnson radius. Future work of interest includes the combination of these results as well as the use of list or interleaved decoders instead of a BMD decoder for the codewords of the constituent code.…”