IERG 6120 Lecture 22 Kenneth Shum Dec 2016

IERG 6120 Lecture 22 Kenneth Shum Dec 2016

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

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. – The code is closed under addition and scalar multiplication • Dimension k : = log(M)/log(q). Dec 2016 kshum 3

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. Dec 2016 kshum 4

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. Dec 2016 kshum 5

Generator matrix of an MDS code • Every k k submatrix is nonsingular • It requires that n!/((n-k)!k!) determinants are nonzero. Dec 2016 kshum 6

Generator matrix of a systematic MDS code Every square submatrix is nonsingular • No. of conditions = Dec 2016 kshum 7

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. Dec 2016 kshum 8

Locality constraint • Example: a code of length 2 m – Codeword (c 1 c 2 c 3 … cm cm+1 cm+2 … c 2 m) – (local parity) cm = c 1 +c 2 +c 3 +… + cm-1. – (local parity) c 2 m = cm+1 + cm+2 + cm+3 +… + c 2 m-1. – (global parity) c 2 m-1 = a 1 c 1 + a 2 c 2 + … +am-1 cm-1 + am+1 cm+1 + am+2 cm+2 + … + a 2 m-2 c 2 m-2. – ai’s are coefficients to be determined. – Any coded symbol can be repaired by accessing m – 1 other symbols c 1 c 2 … cm-1 cm+1 … cm-2 c 2 m-1 cm Dec 2016 cm+2 kshum 9

An example with m = 3 Recover using the two locality constraints m=3, Hamming distance = 3 c 2 c 1 c 4 c 5 c 3 Dec 2016 c 6 kshum 10

Recover any two erasures Recover using the global parity-check equation m=3, Hamming distance = 3 c 2 c 1 c 4 c 5 c 3 Dec 2016 c 6 kshum 11

Cannot tolerate any three erasures m=3, Hamming distance = 3 It is information-theoretically impossible to recover information symbol c 4. c 2 c 1 c 4 c 5 c 3 Dec 2016 c 6 kshum 12

Cannot tolerate any three erasures It is information-theoretically impossible to recover information symbol c 1 and c 2 c 1 c 4 c 5 c 3 Dec 2016 c 6 kshum 13

Maximal recoverability However, it is possible to choose the coefficients ai such that any other combinations of three erasures are recoverable. c 2 c 1 c 4 c 5 c 3 Dec 2016 c 6 kshum 14

Signal flow graph A vertex is associated with a variable. An edge is associated with a coding coefficient. 1 1 1 c 2 c 4 1 a 1 1 a 2 a 3 Source symbols Dec 2016 1 kshum c 2 code symbols c 1 c 3 c 4 c 5 1 c 6 15

Linkage Given a digraph G, and a subset of vertices B c 1 c 2 c 1 c 3 c 2 c 4 c 5 B Dec 2016 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. c 6 kshum 16

Linkage (cont’d) Given a digraph G, and a subset of vertices B c 1 c 2 c 1 c 3 c 2 c 4 c 5 B Dec 2016 Example: {c 1, c 2, c 3} cannot be linked from B. Example: {c 4, c 5, c 6} cannot be linked from B. c 6 kshum 17

Separator c 1 B c 2 c 1 c 3 c 2 c 4 A c 5 A separator consisting of c 4 and c 5. Dec 2016 A subset C of vertices separates A and B if every path from B to A intersect C. c 6 kshum 18

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. Dec 2016 kshum 19

An example of separator c 1 B A c 2 c 1 c 3 c 2 c 4 c 5 c 6 Dec 2016 kshum 20

A example of separator of size 2 c 1 B c 2 c 1 c 3 c 2 c 4 c 5 A c 6 Dec 2016 kshum 21

An example of separator of size 3 c 1 B c 2 c 1 c 3 c 2 A c 4 c 5 c 6 Dec 2016 kshum 22

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. Dec 2016 kshum 23

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 theory of network coding, this is essentially the same as a generic network code. Dec 2016 kshum 24

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. 2016. • D. J. A. Welsh, Matroid theory, Dover 1976. Dec 2016 kshum 25