21 research outputs found
Construction of Partial MDS and Sector-Disk Codes With Two Global Parity Symbols
Partial MDS (PMDS) codes are erasure codes combining local (row) correction with global additional correction of entries, while sector-disk (SD) codes are erasure codes that address the mixed failure mode of current redundant arrays of independent disk (RAID) systems. It has been an open problem to construct general codes that have the PMDS and the SD properties, and previous work has relied on Monte-Carlo searches. In this paper, we present a general construction that addresses the case of any number of failed disks and in addition, two erased sectors. The construction requires a modest field size. This result generalizes previous constructions extending RAID 5 and RAID 6
Construction of Partial MDS and Sector-Disk Codes With Two Global Parity Symbols
Partial MDS (PMDS) codes are erasure codes combining local (row) correction with global additional correction of entries, while sector-disk (SD) codes are erasure codes that address the mixed failure mode of current redundant arrays of independent disk (RAID) systems. It has been an open problem to construct general codes that have the PMDS and the SD properties, and previous work has relied on Monte-Carlo searches. In this paper, we present a general construction that addresses the case of any number of failed disks and in addition, two erased sectors. The construction requires a modest field size. This result generalizes previous constructions extending RAID 5 and RAID 6
Optimal Rebuilding of Multiple Erasures in MDS Codes
MDS array codes are widely used in storage systems due to their
computationally efficient encoding and decoding procedures. An MDS code with
redundancy nodes can correct any node erasures by accessing all the
remaining information in the surviving nodes. However, in practice,
erasures is a more likely failure event, for . Hence, a natural
question is how much information do we need to access in order to rebuild
storage nodes? We define the rebuilding ratio as the fraction of remaining
information accessed during the rebuilding of erasures. In our previous
work we constructed MDS codes, called zigzag codes, that achieve the optimal
rebuilding ratio of for the rebuilding of any systematic node when ,
however, all the information needs to be accessed for the rebuilding of the
parity node erasure.
The (normalized) repair bandwidth is defined as the fraction of information
transmitted from the remaining nodes during the rebuilding process. For codes
that are not necessarily MDS, Dimakis et al. proposed the regenerating codes
framework where any erasures can be corrected by accessing some of the
remaining information, and any erasure can be rebuilt from some subsets
of surviving nodes with optimal repair bandwidth.
In this work, we study 3 questions on rebuilding of codes: (i) We show a
fundamental trade-off between the storage size of the node and the repair
bandwidth similar to the regenerating codes framework, and show that zigzag
codes achieve the optimal rebuilding ratio of for MDS codes, for any
. (ii) We construct systematic codes that achieve optimal
rebuilding ratio of , for any systematic or parity node erasure. (iii) We
present error correction algorithms for zigzag codes, and in particular
demonstrate how these codes can be corrected beyond their minimum Hamming
distances.Comment: There is an overlap of this work with our two previous submissions:
Zigzag Codes: MDS Array Codes with Optimal Rebuilding; On Codes for Optimal
Rebuilding Access. arXiv admin note: text overlap with arXiv:1112.037