More Codes Never Enough

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More Codes Never Enough
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EVENODD Code

• Basics of EVENODD code
• each storage node as a single column
• of data nodes k p (prime)
• of total nodes n p2
• encoding and decoding
• pure XOR operations
• MDS property (r 2)
• tolerate any 2 node failures

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EVENODD Code Encoding
parity I

• Parity node I
• Simple horizontal parity
• Parity node II
• Diagonal parity with adjuster complement

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EVENODD Code Encoding
parity I
parity II









adjuster

• Parity node I
• Simple horizontal parity
• Parity node II
• Diagonal parity with adjuster complement

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EVENODD Encoding

• Numerical example

parity
data
0 0 0 1 0
1 1 0 0 0
0 1 0 0 0
1 1 0 1 1
1

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EVENODD Code Decoding
node failures

















adjuster

• Zig-Zag decode algorithm
• Recover adjuster
• Find a start point
• Decode iteratively

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r 3
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STAR Code
parity III

• 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

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STAR Code Decoding
node failures

• 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

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STAR Code Decoding (cont.)
node failures

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

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Comparison with Extended EVENODD Code

• Similarities
• pure XOR-based
• (k3, 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

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Decoding Complexity
STAR vs. Extended EVENODD
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Decoding Performance

• per node 2880 byte, XOR-based RS implementation
  from J. Blomer

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Bit-Decoding
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Bit-Decoding of EVENODD
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Bit-Decoding of EVENODD
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Bit-Decoding of EVENODD
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Bit-Decoding of EVENODD
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Optimal Updates
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More on EVENODD

• Encoding Complexity
• Decoding Complexity
• Update Complexity

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Update Complexity

• EVENODD 3 2/p
• Lower Bound?

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Update Complexity

• EVENODD 3 2/p
• Lower Bound 2 1/p
• Gap 49

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EVENODD-2?

• Update Complexity 2 1/p
• May be extended to r 3
• r gt 4?