1. Erasure Correcting Codes for Highly Available Storage
Thomas Schwarz, S.J.

2. Error Control Codes Use redundancy to correct errors Designed for
Ease of Encoding
Decoding (Calculation of syndrome / location of error)
Error Correction Power (Burst Errors / Low Redundancy)

3. Error Control Codes Block Codes: Information Symbols + Parity Symbols
(i1i2 i3 i4 i5 i 6 i7 i8 p1 p2 p3)

4. Error Control Codes Typical Applications: Communication:
Deep Space “A match made in heaven”
Telephone
Computer Networks
Streaming Audio, Video (CD, DVD)
Storage (Main Memory, Magnetic & Optical Devices)

5. Error Correcting Codes
Most applications use hardware implemented encoding and decoding.

6. Erasure Correcting Codes
Protect against erasure of data.
Simplest Erasure Correcting Code: Parity
i1 i2 i3 i4 i5 i6 i7 i8 p where p = i1 i2  i3  i4  i5  i6  i7  i8

7. Erasure Correcting Codes
Some applications implement encoding and decoding in hardware (e.g. RAIDs).
Software implementation is much more feasible because of the simpler decoding problem.

8. Erasure Correcting Codes
Ideal Properties:
Systematic: Data is stored explicitly. Data updates do not change other data.
MDS: Only as much parity data is created as is necessary to reconstruct maximum level of failures
Simple encoding and decoding.

9. Parity Based Codes
Only use parity of data (XOR operation) for ease of coding and decoding.

10. Parity Based Codes
History: Protection for Multitrack Magnetic Recording.
Prusinkiewicz & Budkowski 1976:
X X X X X X X X X X
Parity 1
X X X X X X X X X X
Data 1
X X X X X X X X X X
Data 2
X X X X X X X X X X
Data 3
X X X X X X X X X X
Parity 2
Horizontal and diagonal parity.

11. Parity Based Codes
Extend the scheme by using lines of different slopes.
Patel 1985: horizontal + 2 diagonals (slopes 0,1,-1)
However, the code is optimal only if the data band is infinite.
If not, there is (slightly) more parity than data.

12. Parity Based Array Codes
Idea: Break up data into m symbols. Arrange the symbols in columns. Use horizontal and vertical lines to calculate parity.
1st column: horizontal parity, 2nd column: vertical parity

13. Parity Based Array Codes
But is it not so simple!
Is a legitimate code word.

14. Parity Based Array Codes
But indistinguishable from the zero code word after failure of columns 1 and 3.

15. Parity Based Array Codes
Number of Data Columns needs to be prime.

16. EvenOdd Better version of array codes for two parity
Code words two-dimensional m-1 by m arrays with two additional parity columns

17. EvenOdd
The EvenOdd code has as code words the m-1 by m+2 array of symbols ai,j such that

18. EvenOdd Encoding
Set m=5. Start with an arbitrary 4 by 5 data array.

19. EvenOdd Encoding Fill in the horizontal parity lines:
and calculate S to be a3,1+a2,2+a1,3+a0,4
S= = 1.

20. EvenOdd Encoding

21. EvenOdd Decoding
Assume that the last two data columns have failed.

22. EvenOdd Decoding
Use the parity columns to calculate S.

23. EvenOdd Decoding
Use S=1 and the magenta diagonal to find the data symbol in the last column.

24. EvenOdd Decoding
Then use the horizontal parity for one more symbol.

25. EvenOdd Decoding
The blue diagonal now can be exploited.

26. EvenOdd EvenOdd requires m is a prime.
Hence, for a given number n of data lines, choose m to be the smallest prime  n.
Set the superfluous data columns to zero:

27. EvenOdd Encoding and Decoding only uses XOR operations.
Given formulae suggests an iterative procedure, but the equations can be easily expanded to calculate the symbols in parallel.

28. Higher Array Codes
There exists array codes using only XOR operations that can correct up to m erasures.
The decoding process involves solution of a linear equation.

29. Algebraic Block Codes
Interpret symbols (larger than bits) as elements of a Galois Field.
Calculate parity symbols as linear combinations of the data symbols.

30. Galois Fields Only GF(2f) for simplicity’s sake.
Elements: Bit strings of length f.
Addition: XOR
Multiplication: Much more complicated.

31. Galois Field Multiplication
For GF(28). Elements are bytes.
Method 1: Identify byte with a binary polynomial.
E.g. ( ) = x6+x3+1
Multiply to polynomials as polynomials modulo a generator polynomial.
E.g. modulo = x8+x4+x3+x2+1.

32. Galois Field Multiplication
Combination of XORs and shifts!

33. Galois Field Multiplication
This multiplication gives a field structure to GF(2f).
Multiplicative group is cyclic:
There are elements  such that all nonzero elements can be written as
i , i=0,1 … 2f-1.

34. Galois Field Multiplication
For each non-zero element x  GF(2f) define
log(x)=i iff i=x.
Define
antilog(i) = i
Calculate
xy = antilog(log(x)+log(y)); if x0y
= 0; if x=0 or y=0.

35. Galois Field Multiplication
Can be implemented with
two tables,
two zero comparisons,
four additions
three memory accesses.
9 elementary operations in a processor with sufficient L1 cache to store 3*(2f –1) entries.

36. Linear Erasure Correcting Block Codes
m data symbols u = (u0,u1,u2…um-1) u0
u0’
u0’’
u0’’’ . u1
u1’
u1’’
u1’’’ . u2
u2’
u2’’
u2’’’ . u3
u3’
u3’’
u3’’’ .
Code Word u’’
Bucket 0
Bucket 3

37. Linear Erasure Correcting Block Codes
Add k=n – m parity symbols for code word a u0
u0’
u0’’
u0’’’ . u1
u1’
u1’’
u1’’’ . u2
u2’
u2’’
u2’’’ . u3
u3’
u3’’
u3’’’ . p0
p0’
p0 ’’
p0’’’ .
pk-1
pk -1’
pk -1’’
pk-1’’’ .
Parity Bucket k-1
Bucket 0
Bucket 3

38. Linear Erasure Correcting Block Codes
Calculate the parity symbols as a linear combination of the data symbols:
With “Generator Matrix” G.

39. Properties of a Good Generator Matrix
Systematic: Left m by m matrix is identity matrix.
MDS: All matrices formed from m different columns of G are invertible Thus: Any m coordinates of code word a suffice to calculate data word u.

40. Generation of Generator Matrices
Find the largest rectangular matrix with MDS property.
Multiply from left with the inverse of the matrix formed by the first m columns. Result is still MDS and now systematic.

41. Large MDS Matrices
There are known families of matrices with the MDS property:
Cauchy m+n = 2f
Vandermonde n=2f–1
Twice extended Vandermonde n =2f+1

42. Vandermonde Matrix

43. Vandermonde Generator Matrix

44. Vandermonde Generator Matrix
Write column m as a linear combination of the first m columns.
Multiply column i (i=0,1,…m – 1) with this coefficient (non-zero according to Cramer’s Rule. (This preserves MDS.)
Multiply with A-1, where A is the matrix consisting of columns 0 to m – 1.

45. Vandermonde Generator Matrix

46. RS Erasure Correcting Codes
The generator matrix is that of a twice extended, generalized Reed-Solomon code.
Large number of parity symbols:
If symbols are bytes, then code length is 257.

47. RS Erasure Correcting Codes
Encoding:
Generation of a parity symbol costs:
m multiplications with known coefficients
m-1 XOR operation
7m-1 elementary operations

48. RS Erasure Correcting Codes
Change of one data symbol in a data word:
Calculate the difference d = uinew – uinew.
Send d to the site maintaining the parity symbol.
Multiply with coefficient gi,l of G.
Add to existing parity.
7 elementary operations per parity site.
1 elementary operation at data site.
1 message.

49. RS Erasure Correcting Codes
Erasure Correction:
Typical cases:
Parity site has failed. Regenerate parity from the data sites.
Data site has failed. Use column m to regenerate the data from the other data sites and the XOR stored at this first parity site.

50. RS Erasure Correcting Codes
Erasure Correction General Case:
Collect m survivors among data and parity sites
Invert the matrix consisting of the corresponding columns of G
Each replacement site uses this matrix and G in order to calculate a decoding matrix H

51. RS Erasure Correcting Codes
Send surviving data to all replacement sites.
Use decoding matrix in order to regenerate the lost data or parity.

52. Measurements XOR Update: 0.45 sec EvenOdd Update: 0.48 sec
RS Update: 1.27 sec
Record Group with 4 records of 100 bytes.
One record is changed. Measured is time to update parity symbol.
Used 700 MHz Pentium 3 Machine.