# A new sub-packetization bound for minimum storage regenerating codes

Journal Article

Codes for distributed storage systems are often designed to sustain failure of multiple storage disks. Specifically, an (n, k) MDS code stores k symbols in n disks such that the overall system is tolerant to a failure of up to n - k disks. However, access to at least k disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length ℓ. MDS array codes can potentially repair a single erasure using a fraction l/(n - k) of data stored in the surviving nodes. We ask the following question: for a given (n, k), what is the minimum vector-length or sub-packetization factor ℓ required to achieve this optimal fraction? For exact recovery of systematic disks in an MDS code of low redundancy, i.e. k/n > 1/2, the best known explicit codes [1] have a sub-packetization factor I which is exponential in k. It has been conjectured [2] that for a fixed number of parity nodes, it is in fact necessary for ℓ to be exponential in k. In this paper, we provide new converse bounds on k for a given ℓ We prove that k ≤ ℓ2 for an arbitrary but fixed number of parity nodes r = n ™ k. For the practical case of 2 parity nodes, we prove a stronger result that k ≤ 4ℓ. © 2013 IEEE.

### Full Text

### Duke Authors

### Cited Authors

- Goparaju, S; Calderbank, R

### Published Date

- December 19, 2013

### Published In

### Start / End Page

- 1616 - 1620

### International Standard Serial Number (ISSN)

- 2157-8095

### Digital Object Identifier (DOI)

- 10.1109/ISIT.2013.6620500

### Citation Source

- Scopus