Bringing Reliability to the Cloud: Regenerating Codes for
Distributed Storage
Nihar B. Shah1 , K.V. Rashmi 1 , P. Vijay Kumar1 , Kannan
Ramchandran2
1 Indian
Institute Of Science, Bangalore,
2
University of California, Berkeley
Chinese University of Hong Kong
Hong Kong, Sep. 21, 2010
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
1/43
Outline
1
Distributed Data Storage
2
Regenerating Codes
Network Coding
3
Results in Perspective
4
Constructions
MBR Code with d = n − 1
MISER Code
The Product-Matrix Code Construction
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
2/43
Outline
1
Distributed Data Storage
2
Regenerating Codes
Network Coding
3
Results in Perspective
4
Constructions
MBR Code with d = n − 1
MISER Code
The Product-Matrix Code Construction
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
3/43
Distributed Data Storage Network
Information pertaining to
a file is dispersed across
data centers in the
network
Server Node
user taps into surrounding
servers to retrieve data
User Node
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
3/43
Network Assumptions
Nodes are prone to failure (nodes down for maintenance, nodes coming
and going)
Would like to minimize the total amount of storage for a desired level
of performance
Desirable to avoid congestion across links in the network
Restrict attention to linear operations at the individual nodes
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
4/43
An Example Node: A Large Data Center
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
5/43
File Size and Alphabet
Data file B over an alphabet A of of size q
Eg.
A = Fq for example
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
6/43
Single-Server Option
All data hosted on a single
server
vulnerable to even a single
node failure
Server Node
links surrounding the
server node will be subject
to congestion
User Node
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
7/43
Replicated Server Option
Data replicated across ` nodes
in the network
But stores `B units of
data
Links surrounding each
server may still be
congested
Server Node
User Node
Shah, Rashmi, Kumar and Ramchandran
Can tolerate ` − 1 node
failures
Regenerating Codes for Distributed Storage Networks
8/43
Erasure Code Option
Data split into k fragments and an
[n, k] MDS code used to generate
n fragments which are stored in n
nodes in the network
each fragment represents a single
symbol from the finite field Fq
Stores kn B units of data
Server Node
User Node
Shah, Rashmi, Kumar and Ramchandran
Links are less likely now to be
congested as data is dispersed
Can tolerate n − k node failures –
greater probability of network
survival
Regenerating Codes for Distributed Storage Networks
9/43
RAID Example
A
B
A
Disk 1
B
Disk 2
A+B
Disk 3
A+θB
Disk 4
each node stores one symbol
then data stored across the
network looks like a codeword
dispersed in space
(4, 2) MDS code
Used in RAID 6
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
10/43
Outline
1
Distributed Data Storage
2
Regenerating Codes
Network Coding
3
Results in Perspective
4
Constructions
MBR Code with d = n − 1
MISER Code
The Product-Matrix Code Construction
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
11/43
But How Does One Handle Node Failure ?
The obvious option:
A
Disk 1
Connect to any k nodes,
A
B
reconstruct entire data file,
New disk 1
B
B
Disk 2
A+B
A+B
then reconstruct data stored in the node
Disk 3
A+θB
Disk 4
(4, 2) MDS code
Used in RAID 6
But downloading B units of data to revive a node that stores B/k units of
data is wasteful!
Is there a better option?
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
11/43
Yes!
Introducing Regenerating Codes
A1
Disk 1
A2
A1
B1
A2
B2
B1
B2
2A1+2A2+B1
A2+2B1+2B2
2A1+4A2+2B1
A2+2B1+4B2
Disk 2
Disk 3
Disk 4
Store vectors as opposed to scalars at
each node and use vector MDS codes.
New alphabet:
A = Fαq , α > 1.
(in the figure, α = 2 and q = 5)
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
12/43
How Does a Vector Alphabet Help?
We illustrate with an example:
A1
A1
A2
A2
Disk 1
B2
1
+B
2
1
2A1+2A2+B1
A2+2B1+2B2
1
+2
A
B2
Disk 2
2 +2B
B1
2A
A2
B1
B1
2A
1 +4A
A1
Disk 3
2A1+4A2+2B1
A2+2B1+4B2
Disk 4
(download 3 half-symbols as opposed to 2 full-symbols)
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
13/43
Fractional Information Transfer
node 1
new
node 1
node 2
node 3
Fractional transfer
on each link
node k
node k+1
node n-1
the key point is that when each
node stores a vector with α
components, it is possible to
transfer a fraction of the information stored in a node to aid
in node repair.
node n
The potential savings can be quantified using principles of network coding
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
14/43
Recap On Notation
{Fq , [n, k], [α, B]}
node 1
node 2
node 3
DC
data
collector
node k
node k+1
n = number of nodes in the
network
k= number of nodes to connect
to for reconstructing file B
α = number of symbols stored per
node
B is the file size
node n
α capacity
nodes
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
15/43
Two Additional Parameters: (d, β)
node 1
node 2
node 3
new
node 1
β
β
α capacity
node
β
β
β
d = number of nodes to
connect to for
regeneration of a failed
node (k ≤ d ≤ n − 1)
β = number of symbols
downloaded from each
node for regeneration
(β < α)
node k
node k+1
node d+1
the quantity dβ is termed
the “repair bandwidth”
node n
α capacity
nodes
Extended Parameter Set: {Fq , [n, k, d], [α, β, B]}
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
16/43
Network Coding Example
x
both sinks want all the
data – an example of a
“multicast”
y
S
x
min-cut capacity to each
sink = 2
y
a
b
y
x
MFMC theorem of
commodity flow tells us
each sink can get the
information
c
x+y
x
y
network coding tells us
that in this situation, both
sinks can recover the
information!
d
x+y
x+y
t1
x
t2
y
Shah, Rashmi, Kumar and Ramchandran
x
y
an example solution is
illustrated
Regenerating Codes for Distributed Storage Networks
17/43
Cut-Set Bound for Regenerating Codes
The network here is the original set of n nodes plus all subsequent
replacement nodes. Turns out, the min-cut capacity is given by:
k−1
X
min{α, (d − i)β} ≥ B
i=0
DC
in
out
cut
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
18/43
The Minimum-Storage Regeneration (MSR) Point
(minimize α, then β)
B ≤
k−1
X
min{α, (d − i)β}
i=0
B =
k−1
X
α,
(d − i)β ≥ α,
B
k
β=
all i
i=0
i.e., α =
or, α = β(d − k + 1),
α
d −k +1
B = αk
(Note: both α, B are multiples of β)
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
19/43
The Minimum-Bandwidth Regeneration (MBR) Point
(minimize β, then α)
B ≤
k−1
X
min{α, (d − i)β}
i=0
B =
k−1
X
(d − i)β,
α ≥ (d − i)β, all i
i=0
k
i.e., B = [dk −
]β
2
α = dβ
(Note: again, both α, B are multiples of β)
(other operating points are possible; in general, there is a tradeoff)
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
20/43
The Storage-Repair Bandwidth Tradeoff Curve
Repair bandwidth dβ versus storage α
MSR and MBR are the two extreme points
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
21/43
Summarizing MSR and MBR Parameters
(storage per node, repair bandwidth)
MSR Point
αmin
B
B
=
, dβmin =
k
k
(store least possible;
1
1 − ( k−1
d )
!!
download a little more)
MBR Point
αmin
B
=
k
!
1
1−
(store a little more;
Shah, Rashmi, Kumar and Ramchandran
(k−1)
2d
, dβmin
B
=
k
1
1−
!!
(k−1)
2d
download only what you store)
Regenerating Codes for Distributed Storage Networks
22/43
β = 1 and “Striping”
at both MSR, MBR endpoints, α, B are multiples of β
β dictates smallest size of file for which a solution exists
a solution for β = 1 can be replicated to give solutions for β > 1
(“striping” of data)
thus constructions for β = 1 are of greatest interest
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
23/43
Exact versus Functional Regeneration
B ≤
k−1
X
min{α, (d − i)β}
i=0
Bound is known to be achievable for functional regeneration
What if one demanded exact regeneration ?
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
24/43
Outline
1
Distributed Data Storage
2
Regenerating Codes
Network Coding
3
Results in Perspective
4
Constructions
MBR Code with d = n − 1
MISER Code
The Product-Matrix Code Construction
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
25/43
Brief History
Dimakis et al. 2007, Wu 2009
Existence: functional regenerating codes.
1
Wu et al. 2009
E-MSR
[n, 2, n
n-1]
1]
IA
Our group 2009
E-MBR
E
MBR (uncoded repair)
[n, k, n-1]
2
AE-MSR
[ k,
[n,
k kk+1]
1]
3
Cullina et al. 2009
E-MSR
[5, 3, 4], [4, 2, 3]
Computer search
4
5
LEGEND
[*, *, *]: [n, k, d]
: Explicit codes
E : Exact
AE : Approx. exact
SE : Exact, only sys. nodes
IA : Interference alignment
Shah, Rashmi, Kumar and Ramchandran
Wu 2009
S
AE-MSR
[n ≥2k, k, k+1]
Our group 2009
SE-MSR
[n ≥2k, k, n-1],
non-ach of MSR
d < 2k-3, β = 1
IA
6
Regenerating Codes for Distributed Storage Networks
25/43
Brief History (2)
Our group 2010
SE-MBR
all [n, k, d]
Non-ach
Non
ach. of
interior points
Suh et al. 2009
Show regeneration of
non-sys. nodes for code in (5)
IA
7
8
Our group 2010
Flexible class of regenerating codes
9
Cadambe et al. 2010
E-MSR
[n k,
[n,
k nn-1]
1], B →∞
IA
LEGEND
[*, *, *]: [n, k, d]
: Explicit codes
E : Exact
AE : Approx. exact
SE : Exact, only sys. nodes
IA : Interference alignment
Shah, Rashmi, Kumar and Ramchandran
10
11
Our group 2010
E-MSR
[n, k, d ≥2k-2],
E-MBR
[n, k, d]
Product Matrix
Suh et al. 2010
E-MSR
[n, k, d] , B →∞
IA
12
Regenerating Codes for Distributed Storage Networks
26/43
Our Contributions
Constructions
E-MBR code d = n − 1
AE-MSR code d = k + 1
SE-MSR code, d = n − 1 exact regeneration of systematic nodes
Most recently, a unified product-matrix construction for MSR, MBR, all
d (essentially)
Non-achievability of interior points under exact regeneration
non-existence of MSR code with d < 2k − 3 under exact regeneration
when β = 1
the cut-set bound, existence of codes and a preliminary construction
for a more flexible regeneration code set up
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
27/43
Outline
1
Distributed Data Storage
2
Regenerating Codes
Network Coding
3
Results in Perspective
4
Constructions
MBR Code with d = n − 1
MISER Code
The Product-Matrix Code Construction
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
28/43
An Example Construction for the MBR d = n − 1 Point
n = 5, k = 3, d = 4
B = 9, α = 4, β = 1
Gen Mx [10, 9] MDS Code:
mt [v 1 v 2 · · · v 10 ]
node 1
v1 v2 v3 v4
node 1
node 2
node 3
v1 v5 v6 v7
v4
node 4
v3 v6 v8 v10
node 5
v4 v7 v9 v10
v5
v6
v9
v10
node 2
v7
v2 v5 v8 v9
node 5
v1
v2
v3
node 3
v8
node 4
K. V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Explicit Construction of Optimal Exact Regenerating Codes for
Distributed Storage,” in Proc. Allerton Conference on Control, Computing and Communication, Urbana-Champaign, Sep. 2009.
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
28/43
The General Case: MBR d = n − 1 Code
node 1
For the general case, the MDS
code would have
n
length =
2
k
dimension = dk −
2
v4
node 5
v1
v7
v5
v6
v9
v10
node 2
v2
v3
node 3
v8
node 4
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
29/43
MISER Code
Family of MSR codes for d = n − 1 ≥ 2k − 1 that perform
reconstruction
optimal exact-repair of systematic nodes
Based on the concept of Interference Alignment
Suh-Ramchandran show that this code can perform optimal
exact-repair of parity nodes as well
N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran,“Explicit codes minimizing repair bandwidth for distributed
storage,” in Proc. ITW, Cairo, Jan. 2010.
N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran,“Interference Alignment in Regenerating Codes for Distributed
Storage: Necessity and Code Constructions,” submitted to IEEE Trans. on Information Theory, available online at
arXiv:1005.1634v2 [cs.IT].
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
30/43
MISER
MISERCode:
Code: Toy
ToyExample
Example
n = 4, k = 2, d = 3
n = 4, k = 2, d = 3
α = d − k + 1 = 2, B = kα = 4
α = d − k + 1 = 2, B = kα = 4
Field of operation: F5
Field of operation: F5
Node 1
z1
z2
Node 2
z3
z4
Node 3
2z1 + 2z2 + z3
z2 + 2z3 + 2z4
Node 4
2z1 + 4z2 + 2z3
z2 + 2z3 + 4z4
Node 1
y1 + 2y3
y1 + y2 + 4y3 + 2y4
Shah,
2y
+ and
3y
+Ramchandran
3y + y
Shah,Rashmi,
Rashmi,Kumar
Kumar
andRamchandran
z3
New Node 1
z
2z 2 + 3
2z 1 +
2 z 3`
+
+ 4z 2
z
1
2
interference
is aligned
3y
1 +y
Codes
for
StorageStorage
Networks
+3 Distributed
2y + 3yRegenerating
2Codes
Regenerating
for Distributed
31/43
30/30
Product-Matrix Framework
C = |{z}
Ψ |{z}
M
|{z}
n×α
n×d d×α
M : Message matrix
Contains message symbols with some message symbols repeated
Possesses a block-symmetry property
Ψ : Encoding matrix
Used to disperse information across the nodes
Independent of message symbols
C : Code matrix
Each row represents one node
i th node stores: ψ ti M
K. V. Rashmi, N. B. Shah, and P. V. Kumar, “Optimal Exact-Regenerating Codes for the MSR and MBR Points via a
Product-Matrix Construction,” submitted to IEEE Transactions on Information Theory, available online at arxiv:1005.4178
[cs.IT].
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
32/43
The Product-Matrix MBR (PM-MBR) Code
α=d
B = kd −
k
2
→
B=
k+1
2
+ k(d − k)
Let S be a (k × k) symmetric matrix with
symbols
k+1
2
distinct message
Let T be a (k × (d − k)) matrix with k(d − k) distinct message
symbols
thus all message symbols are accounted for
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
33/43
Product-matrix MBR Code

Message matrix



M
=
|{z} 

d×d
S
|{z}
T
|{z}
k×k
k×(d−k)
Tt
|{z}
0
|{z}
(d−k)×k
Encoding matrix
Ψ =
|{z}
n×d
Φ
|{z}
n×k






(symmetric)
k×(d−k)
∆
|{z}
n×(d−k)
Φ: any k rows linearly independent
Ψ: any d rows linearly independent
e.g., Cauchy, Vandermonde matrix
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
34/43
Product-matrix MBR Code : Data Reconstruction
Node i passes: ψ ti M

Aggregator y
ΨDC M
(ΨDC = [ΦDC ∆DC ] is (k × d))

y
M=
Decoder
ΦDC S + ∆DC
Tt
ΦDC T
↓
Ψ=
S
Tt
Φ
T
0
∆
C = ΨM
ΦDC is k × k, invertible
Decode T
↓
Subtract ∆DC T t , Decode S
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
35/43
Product-matrix MBR Code : Exact Regeneration
Replacement node f needs: ψ tf M
Helper node i, 1 ≤ i ≤ d stores: ψ ti M
————————————————————
Helper node i passes: ψ ti Mψ f

Aggregator y
Ψrepair Mψ f
(Ψrepair is d × d, invertible)

Partial Decoder y
Mψ f
M=
Ψ=
S
Tt
Φ
T
0
∆
C = ΨM
(M is symmetric)

Re-encoder y
ψ tf M
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
36/43
The Product-matrix MSR Code Parameters
Here again β = 1
α=d −k +1
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
37/43
The MSR point-Numerology
d < 2k − 3 not possible with β = 1
This code is designed for d ≥ 2k − 2
Choose d = 2k − 2 first, then extend to higher d
Gives
k = α+1
d
= 2α
B = α(α + 1)
S1 , S2 : (α × α) symmetric matrices with
symbols each
Shah, Rashmi, Kumar and Ramchandran
α(α+1)
2
distinct message
Regenerating Codes for Distributed Storage Networks
38/43
The Product-Matrix MSR Code

Message matrix


M
=
|{z} 

d×α
S1
|{z}
α×α
S2
|{z}





α×α
Encoding matrix
Ψ =
|{z}
n×d
Φ
|{z}
n×α
ΛΦ
|{z}
n×α
Φ: any α rows linearly independent
Λ: n × n diagonal matrix with the diagonal elements distinct
Ψ: any d rows linearly independent
e.g., Vandermonde
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
39/43
The Product-Matrix MSR Code-Data Reconstruction
Node i passes: ψ ti M

Aggregatory
ΨDC M
(ΨDC = [ΦDC ΛDC ΦDC ] is k × d)
↓
[ΦDC S1 + ΛDC ΦDC S2 ]

Decodery
[ΦDC S1 ΦtDC + ΛDC ΦDC S2 ΦtDC ]
↓
[P + ΛDC Q]
(P and Q symmetric)
M=
Ψ=
S1
S2
Φ
ΛΦ
C = ΨM
↓
(i, j) : Pij + λi Qij , (j, i) : Pij + λjQij
(Solve for P and Q)
↓
Recover S1 and S2
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
40/43
The Product-Matrix MSR Code-Exact Regeneration
Replacement node f needs: ψ tf M
Helper node i stores: ψ ti M
————————————————————
Helper node i passes: ψ ti Mφf

Aggregator y
Ψrep Mφf
(Ψrep is d × d, invertible)

Partial Decoder y
S1 φf
Mφf =
S2 φf

Re-encoder y
M=
Ψ=
S1
S2
Φ
ΛΦ
C = ΨM
φtf S1 + λf φtf S2 = ψ tf M
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
41/43
References
A. G. Dimakis, P. B. Godfrey, Y. Wu, M. Wainwright, and K. Ramchandran, “Network Coding for Distributed Storage
Systems,” to appear in IEEE Transactions on Information Theory, available online at arXiv:0803.0632 [cs.IT].
Y. Wu and A. Dimakis, “Reducing Repair Traffic for Erasure Coding-Based Storage via Interference Alignment,” in Proc.
IEEE International Symposium on Information Theory, Jul. 2009.
Y. Wu, “Existence and Construction of Capacity-Achieving Network Codes for Distributed Storage,” in Proc. IEEE
International Symposium on Information Theory, Seoul, Jul. 2009.
K. V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Explicit Construction of Optimal Exact Regenerating
Codes for Distributed Storage,” in Proc. Allerton Conference on Control, Computing and Communication,
Urbana-Champaign, Sep. 2009.
D. Cullina, A. G. Dimakis and Tracey Ho, “Searching for Minimum Storage Regenerating Codes,” in Proc. Allerton
Conference on Control, Computing and Communication, Urbana-Champaign, September 2009.
N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran,“Explicit Codes Minimizing Repair Bandwidth for
Distributed Storage,” in Proc. IEEE Information Theory Workshop, Cairo, Jan. 2010.
Y. Wu, “A Construction of Systematic MDS Codes with Minimum Repair Bandwidth,” submitted to IEEE Transactions
on Information Theory, available online at arXiv:0910.2486v1 [cs.IT].
K. V. Rashmi, N. B. Shah and P. V. Kumar, “Optimal Exact-Regenerating Codes for the MSR and MBR Points via a
Product-Matrix Construction,” submitted to IEEE Transactions on Information Theory, available online at
arxiv:1005.4178 [cs.IT].
N. B. Shah, K. V. Rashmi, and P. V. Kumar “A Flexible Class of Regenerating Codes for Distributed Storage,” in Proc.
IEEE International Symposium on Information Theory, Austin, Jun. 2010.
K. V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran “Explicit and Optimal Exact-Regenerating Codes for the
Minimum-Bandwidth Point in Distributed Storage,” in Proc. IEEE International Symposium on Information Theory,
Austin, Jun. 2010.
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
42/43
References
C. Suh and K. Ramchandran, “Interference Alignment Based Exact Regeneration Codes for Distributed Storage,” in Proc.
IEEE International Symposium on Information Theory, Austin, Jun. 2010.
S. Pawar, S. El Rouayheb and K. Ramchandran, “On Secure Distributed Data Storage Under Repair Dynamics,” in Proc.
IEEE International Symposium on Information Theory, Austin, Jun. 2010.
V. R. Cadambe, S. A. Jafar, and H. Maleki, “Distributed Data Storage with Minimum Storage Regenerating Codes Exact and Functional Repair are Asymptotically Equally Efficient,” available online at arXiv:1004.4299v1 [cs.IT].
C. Suh and K. Ramchandran, “On the Existence of Optimal Exact-Repair MDS Codes for Distributed Storage,” available
online at arXiv:1004.4663v1 [cs.IT].
Dennis S. Bernstein, Matrix mathematics: Theory, facts, and formulas with application to linear systems theory, Princeton
University Press, Princeton, NJ, p.119, 2005.
S. Rhea, P. Eaton, D. Geels, H. Weatherspoon, B. Zhao, and J. Kubiatowicz, “Pond:the OceanStore prototype,” in Proc.
USENIXFile and Storage Technologies (FAST), 2003.
R. Bhagwan, K. Tati, Yu Chung Cheng, S. Savage, and G. M. Voelker, “Total recall: System support for automated
availability management,” in NSDI, 2004.
R. Koetter and M. Medard, “An algebraic approach to network coding,” IEEE/ACM Transactions on Networking, v.11
n.5, p.782-795, Oct. 2003.
Thanks!
Shah, Rashmi, Kumar and Ramchandran
Regenerating Codes for Distributed Storage Networks
43/43