Parity Augmentation

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Parity Augmentation

• An Alternative Approach to LDPC Decoding

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Background

• LDPC codes, developed by Robert Gallager, are
  among the best codes for error correction coding
• However, the decoding process can be long and
  involved as well as take more iterations than are
  feasible for data which needs to be transmitted
  quickly

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Alternative Algorithms

• Sinkhorn is a process in which a matrix is made
  doubly stochastic (all rows and columns sum to 1)
  by repeatedly normalizing both the rows and
  columns
• Sinkhorn worked much better than belief
  propagation when solving Sudokus, so the question
  arose as to if Sinkhorn would surpass Belief
  Propagation for LDPC decoding

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• The key to solving Sudoku with Sinkhorn was Math
  by wish fulfillmentthat is, the matrices were
  assumed to be doubly stochastic and were thus
  altered on those constraints
• Similarly, Parity Augmentation does math by wish
  fulfillment in its . . . It does it some how, I
  think

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Basis behind Parity Augmentation

• When decoding using a parity check matrix H, the
  message is considered decoded if all the parities
  checkthat is, if the decoded message lies in the
  nullspace of H.
• Therefore, by maximizing the probabilities in
  which parities check for each row, eventually
  even parity can be reached

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Equations galore

• By Gallagers lemma, the probability that n bits
  sum to 0 in GF(2) is
• 1pi blahblah thats what LaTeX is for
• Similarly, the probability that n bits sum to 1
  is _____

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Click to add Title

• For a single bit p_i, alter it by multiplying it
  by the probability that the other bits sum to one
  and normalizing by P_E. Notice that if the
  extrinsic probability gt P_E, then the bit is more
  likely to be 1 than 0, since the bit and the sum
  of the other bits must sum to 0.
• Finally, find the new probability of even parity

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• The success of this algorithm is dependent on
  even parity increasing at each succesive step.
  It does.
• Proof

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How does it compare?

• As a rule, Parity Augmentation is faster, less
  complex, and takes fewer iterations to achieve
  success than the Gallager decoder.
• However, Parity Augmentation tends to perform
  slightly worse. This is perhaps because the
  increase in even parity may be miniscule.

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Decoding Performance for ½ code
Notice how the Parity reaches a limit where more
iterations dont improve its performance, but the
Gallager continues to improve with each
successive iteration
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Therefore, what?

• The algorithm generally stopped working when
  there were too many probabilities or they hovered
  around .5, so if that kind of situation could be
  avoided or remedied, the performance would
  improve
• The lemma that took For.Ev.Er to figure out may
  prove useful for other instances when even parity
  is a goal.

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Optimization

• Another possible way to decode is through
  gradient descent, that is, the individual values
  are changed by small values as to prevent too
  large of jumps, but rather steer the
  probabilities in the right direction
• Or, in mathematical terms,

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Results of Gradient Optimization

• Has a slightly worse prbobability of error than
  the Parity or Gallager but is comparable time as
  the Parity Augmentation (in other words, faster
  than the Gallager)
• Takes more iterations than either
• So . . . needs some tweaking.

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Conclusion

• LDPC decoding is an exploding area of study.
  Figuring out fast, efficient, and effective ways
  to decode is quite the way to spend your summer