1. More Codes Never Enough

2. 2 EVENODD Code Basics of EVENODD code  each storage node as a single column # of data nodes k = p (prime) # of total nodes n = p+2  encoding and decoding pure XOR operations  MDS property (r = 2) tolerate any 2 node failures parity nodes data nodes

3. 3 EVENODD Code Encoding Parity node I:  Simple horizontal parity Parity node II:  Diagonal parity with adjuster complement parity Iparity II

4. 4 EVENODD Code Encoding Parity node I:  Simple horizontal parity Parity node II:  Diagonal parity with adjuster complement adjuster parity Iparity II

5. 5 EVENODD Encoding 00010 11000 01000 11011 parity 1 0 1 0 Numerical example 0 0 0 1 1 1 1 1 0 data

6. 6 EVENODD Code Decoding Zig-Zag decode algorithm  Recover adjuster  Find a start point  Decode iteratively adjuster node failures

7. r = 3

8. 8 STAR Code Basics of STAR code  Extension of EVENODD code EVENODD code + 1 additional parity node  An efficient MDS code Tolerating up to 3 node failures (r = 3)  Encoding is straightforward parity III

9. 9 STAR Code Decoding Decode algorithm needs to handle any 3 node failures  Special cases can be handled easily (parity failures) e.g. parity node III among the 3 failures  exact EVENODD decode  Difficult part is to deal with 3 information node failures Key to efficient decoding node failures

10. 10 STAR Code Decoding (cont.) In the 2 nd column, the sum of any pair of cells with stride 3 can be recovered. Starting with the last cell (zero), all cells in the 2 nd column can then be recovered. The remaining problem is to recover 2 node failures  apply EVENODD decoding node failures

11. 11 Comparison with Extended EVENODD Code Similarities  pure XOR-based  (k+3, k) MDS Differences  Extended EVENODD slope 0, 1, 2 generalize to tolerate more than triple failures  STAR slope 0, 1, -1 geometric symmetry  faster decoding

12. 12 Decoding Complexity STAR vs. Extended EVENODD

13. 13 Decoding Performance per node 2880 byte, XOR-based RS implementation from J. Blomer

14. Bit-Decoding

15. 15 Bit-Decoding of EVENODD

16. 16 Bit-Decoding of EVENODD

17. 17 Bit-Decoding of EVENODD

18. 18 Bit-Decoding of EVENODD

19. Optimal Updates

20. 20 More on EVENODD Encoding Complexity Decoding Complexity Update Complexity

21. 21 Update Complexity EVENODD: 3 – 2/p Lower Bound?

22. 22 Update Complexity EVENODD: 3 – 2/p Lower Bound: 2 + 1/p Gap: 49%

23. 23 EVENODD-2? Update Complexity: 2 + 1/p May be extended to r = 3 r > 4?