SCIENCE CHINA Information Sciences, Volume 61, Issue 10: 100304(2018) https://doi.org/10.1007/s11432-018-9516-6

## Code constructions for multi-node exact repair in distributed storage$^\dag$

• AcceptedJul 9, 2018
• PublishedAug 17, 2018
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### Abstract

We study the problem of centralized exact repair of multiple failures in distributed storage. We present constructions that achieve a new set of interior points under exact repair. The constructions build upon the layered code Construction by Tian et al., designed for exact repair of single failure. We firstly improve upon the layered Construction for general system parameters. Then, we extend the improved Construction to support the repair of multiple failures, with varying number of helpers. In particular, for some parameters,we prove the optimality of one point in terms of the storage size and the repair bandwidth for multiple erasures. Finally, considering minimum bandwidth cooperative repair (MBCR) codes as centralized repair codes, we determine explicitly the best achievable region obtained by space-sharing among all known points, including the MBCR point.

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• Figure 1

A repair situation associated to given parameters $s$ and $p$.

• Figure 2

(Color online) Using the MSMR repair property improves upon the layered code repair performance. The exact-repair tradeoff is achieved in [12], with lower bound derived in [21].

• Figure 3

(Color online) Comparing achievable schemes for different scenarios. The dashed curves correspond to points achieved with Construction 1, and the solid curves (with open circles) corresponds to Construction 2, coupled with MSMR repair. The lowest solid curve corresponds to the functional repair tradeoff (1). The dotted line corresponds to space-sharing between MSMR and MBCR points. $(n,k,d,e)$ = (8,6,6,2) in (a); (9,6,7,2) in (b); (10,6,8,2) in (c); (19,16,17,2) in (d); (16,14,14,2) in (e); (24,14,14,2) in (f).

• Figure 4

(Color online) Functional tradeoff and optimal points achieved by Proposition 3.2.

• Figure 5

(Color online) Achievable points by Construction 1 for a $(n,k,d,e)=(17,14,14,3)$ system. The $x$-axis is the normalized storage per node $\bar{\alpha}$ and the $y$-axis is the normalized bandwidth $\bar{\beta}$.

• Table 1   EVENODD code: an example of an MSRcode$^{\rm~a)b)}$
 Symbol $1$ Symbol $2$ Symbol $3$ Symbol $4$ $a$ $c$ $a+c$ $~a+d$ $b$ $d$ $b+d$ $b+c+d$
• Table 2   Summary of cases for which $(\bar{\alpha}_r,\bar{\beta}_r)$ is a corner point in $\mathcal{R}$ $^{\rm~a)}$
 $(\bar{\alpha}_r,~\bar{\beta}_r)$ $k~<~k_{\rm~th}(p)$ $k~\geq~k_{\rm~th}(p)$ $~p~\le~p_{\max}$ $\checkmark$ $\text{\ding{55}}$ $p~>~p_{\max}$ $\checkmark$ $\checkmark$
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