1. IERG6120 Lecture 22
Kenneth Shum
Dec 2016

2. Outline MDS codes Codes with locality constraints Menger’s theorem
Maximally recoverable codes

3. What is a code? A set of strings of over a fixed alphabet A.
(n, M)q code
M codewords.
Each codeword consists of n symbols.
Alphabet size = q
Hamming distance measures the error-correcting capability of a code.
In the context of distributed storage, the n symbols in a codeword are stored in n nodes.
Scalar-linear code
The alphabet is a finite field
Each node stores a finite field element
The code is closed under addition and scalar multiplication
Vector-linear code
The alphabet is a vector space over a finite field
Each node stores a vector of  finite field elements.
Dimension k := log(M)/log(q).
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4. Caveats
Some people include the encoder as part of the definition of a code.
Encoder Enc is a linear mapping from a k-dim vector space to an n-dim vector space.
The codewords are all the possible outputs of the encoder.
An encoder is said to be systematic if the k information symbols appear somewhere in the codeword.
We usually re-order the coded symbols so that the k information appear as the first k symbols.
A systematic code is a code together with a systematic encoder.
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5. MDS code Achieve Singleton bound with equality.
No. of codewords = M = qn – d +1
(d denotes the Hamming distance of a code)
If k = log(M)/log(q) is an integer, then any k coded symbols determines the codeword uniquely.
Tolerate any n-k erasures.
Highest level of erasure-correction capability.
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6. Generator matrix of an MDS code
Every kk submatrix is nonsingular
It requires that n!/((n-k)!k!) determinants are nonzero.
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7. Generator matrix of a systematic MDS code
G=\begin{bmatrix}
1 & 0 & \ldots & 0 &
p_{11}& p_{12} & \ldots & p_{1n} \\
0 & 1 & \ldots & 0 &
p_{21}& p_{22} & \ldots & p_{2n} \\
\vdots & \vdots & \ddots & \vdots &
\vdots & \vdots & \ddots & \vdots \\
0&0&\ldots&1&p_{k1}& p_{k2} & \ldots & p_{kn}
\end{bmatrix}
Every square submatrix
is nonsingular
No. of conditions =
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8. Information set
A set of k coded symbols is said to be an information set if it uniquely determine the codeword.
In an MDS code, any subset of k coded symbols is an information set.
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9. Locality constraint … … Example: a code of length 2m c2 c1
Codeword (c1 c2 c3 … cm cm+1 cm+2 … c2m)
(local parity) cm = c1 +c2 +c3 +… + cm-1.
(local parity) c2m = cm+1 + cm+2 + cm+3 +… + c2m-1.
(global parity) c2m-1 = a1c1 + a2c2 + … +am-1cm-1 + am+1 cm+1 + am+2 cm+2 + … + a2m-2 c2m-2.
ai’s are coefficients to be determined.
Any coded symbol can be repaired by accessing m – 1 other symbols
c2m
c2m-1
cm+1
cm+2
cm-2 … c1 cm c2
cm-1 …
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10. An example with m = 3 c1 c2 c4 c5 c3 c6 m=3, Hamming distance = 3
Recover using the two locality constraints c1 c2 c4 c5 c3 c6
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11. Recover any two erasures
m=3, Hamming distance = 3
Recover using the global parity-check equation c1 c2 c4 c5 c3 c6
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12. Cannot tolerate any three erasures
m=3, Hamming distance = 3
It is information-theoretically impossible to recover information symbol c4. c1 c2 c4 c5 c3 c6
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13. Cannot tolerate any three erasures
It is information-theoretically impossible to recover information symbol c1 and c2. c1 c2 c4 c5 c3 c6
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14. Maximal recoverability
However, it is possible to choose the coefficients ai such that any other combinations of three erasures are recoverable. c1 c2 c4 c5 c3 c6
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15. Signal flow graph c1 c2 c1 c3 c2 c4 c4 c5 c6
A vertex is associated with a variable. An edge is associated with a coding coefficient. c1 1 c2 c1 1 1 c3 a1 a2 a3 c2
code symbols c4 1 c4 c5 1
Source symbols c6
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16. Linkage c1 c2 c1 c3 c2 c4 c4 c5 B c6 Given a digraph G,
and a subset of vertices B c1
A subset A of
vertices is said to be linked from B if there is a set of |A| vertices-disjoint paths from B to A. c2 c1 c3 c2 c4 c4 c5 B c6
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17. Linkage (cont’d) c1 c2 c1 c3 c2 c4 c4 c5 B c6 Given a digraph G,
and a subset of vertices B c1
Example: {c1, c2, c3} cannot be linked
from B. c2 c1 c3 c2 c4 c4
Example: {c4, c5, c6} cannot be linked
from B. c5 B c6
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18. Separator c1 B c2 c1 c3 c2 c4 c4 A c5 c6
A subset C of vertices separates A and B if every path from B to A intersect C. c2 c1 c3 c2 c4
A separator consisting of c4 and c5. c4 A c5 c6
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19. Menger’s Theorem
Theorem: Let G = (V,E) be a digraph, and A and B be subsets of V. If A can be linked from B if and only if there is no vertex subset C which separates A from B with |C| < |A| .
Menger’s theorem is a basic theorem in combinatorial optimization and graph theory. Please see e.g. Chapter 13 in the book “Matroid theory” by Welsh for more details.
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20. An example of separatorc1 B A c2 c1 c3 c2 c4 c4 c5 c6
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21. A example of separator of size 2c1 B B c2 c1 c3 c2 c4 c4 c5 A c6
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22. An example of separator of size 3c1 B c2 c1 c3 c2 A c4 c4 c5 c6
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23. Implication
If a set of code symbols cannot be linked to the set of source symbols, then it is information-theoretically impossible to decode the source symbols from them.
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24. Maximal recoverability
Conversely, if there are k source symbols, we can choose the encoding coefficients such that any set of k coded symbols which can be linked back to the set of source symbols is indeed an information set.
Apply Schwartz-Zippel Lemma
In the theory of network coding, this is essentially the same as a generic network code.
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25. References
P. Gopalan, C. Huang, B. Jenkins and S. Yekhanin, Explicit maximally recoverable codes with locality, IEEE Trans. Inf. Theory, Jun, 2014.
M. Blaum, J. S. Plank, M. Schwartz and E. Yaakobi, Construction of partial MDS and sector-disk codes with two global parity symbols, IEEE Trans. Inf. Theory, May
D. J. A. Welsh, Matroid theory, Dover 1976.
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