Generating Code Representations Suitable for Belief Propagation Decoding

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Generating Code Representations Suitable for
Belief Propagation Decoding

• Jonathan Yedidia
• (Mitsubishi Electric Research Labs)
• Jinghu Chen
• (University of Hawaii)
• Marc Fossorier
• (University of Hawaii)

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Outline
Research

• Motivation
• Generalized Parity Check (GPC) Matrices
• An Algorithm for Transforming GPC Matrices
• Results
• Golay Code
• Euclidean Geometry Code
• Relationship to Generalized Belief Propagation
• Future Improvements

Presentation
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Background

• For block-lengths , belief propagation
  (BP) decoding of irregular LDPC codes approaches
  capacity for BEC (provably) and AWGNC
  (empirically).
• For and
  there is still room for improvement. Whats the
  best way to use BP in this regime?

Luby, Mitzenmacher, Shokrollahi, Spielman
2001 Chung, Forney, Richardson Urbanke 2001
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Using BP on Classical Codes

• Advantages
• Many good classical codes in this regime.
• Plenty of theory to exploit.
• Problem codes not normally defined in terms of
  sparse parity check matrices. Exception one-step
  majority-logic decodable codes.
• Lucas, Fossorier, Kou, Lin, 2000

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Representing a code
(7,4) Hamming code
Parity check matrix
Tanner graph
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Other Representations
Circulant parity check matrix
In general, n-k linearly independent rows.
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Generalized Parity Check Matrices
hidden bits
hidden bits
Symbol bits
Symbol bits
M rows, N columns, n symbol bits, N-k linearly
independent rows.
This particular construction obtained from dual
graph of dual codesee Forney, 2001
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Algorithm for Transforming GPC Matrices

• Input GPC matrix for binary linear block code
• Output GPC matrix for same code with
• Fewer ones per row
• More ones per column
• Fewer or no 4-cycles in corresponding factor
  graph
• Uses new hidden bits that represent parities of
  intersection sets of bits.

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Description of Algorithm

• Step 1 Form sets of bits
• Constraint sets 1,2,3,5, 2,3,4,6, 3,4,5,7
• Intersection sets 2,3, 3,4, 3,5
• Single bit sets 1, 2, 3, 4, 5, 6, 7

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Description of Algorithm (cont.)
Step 2 Organize sets into partially ordered set.
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Description of Algorithm (cont.)
Step 3 Make ordered lists of sub-sets for all
constraint and intersection sets

• 1,2,3,5
• 2,3, 3,5, 1
• 2, 3, 5
• 2,3,4,6
• 2,3, 3,4, 6
• 2, 3, 4
• 3,4,5,7
• 3,4, 3,5, 7
• 3, 4, 5

• 2,3
• 2, 3
• 3,4
• 3, 4
• 3,5
• 3, 5

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Description of Algorithm (cont.)

• Step 4 Break down sets into unions

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Description of Algorithm (cont.)

• Step 5 Convert to output GPC matrix

1 2 3 4 5 6 7 23 34 35
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Features and Problems

• Features
• Groups bits in a sensible way to divide and
  conquer problemfewer ones per row.
• Uses redundant checksmore ones per column.
• We can ensure no 4-cycles.
• Problem
• New hidden bits do not get evidence from
  channelthey may introduce uncertainty.

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Toy example Golay code

• Input 23 by 23 circulant parity check matrix
  with 8 ones per row
• Output 495 by 230 GPC matrix with 3-5 ones per
  row
• Note that complexity of BP scales with number of
  ones in GPC matrix and number of iterations, not
  with MN.

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Multi-step Majority Logic Decodable Codes Based
on Euclidean Geometry (EG)

• Generic structure
• We studied a (n255,k127) EG code
• 255 points, 212555355 lines, 5355 planes
• 4 points in each line, 4 lines in each plane
• Each plane has 20 children 5 sets of 4 parallel
  lines

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GPC Matrices for EG Code

• Input 5355 by 255 matrix with 16 ones per row
  many pairs of rows intersect at 4 bits.
• Output

5355 rows with 5 ones in each
26775 rows with 4 ones in each
No 4-cycles in corresponding factor graph
255 columns with 84 ones in each
5355 columns with 21 ones in each
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Generalized Belief Propagation (GBP)

• Can construct new message-passing algorithms
  for any factor graph using region graphs.
• see Yedidia, Freeman Weiss 2000, 2002
  McEliece Yildirim 2002

A
C
B
2
3
1
4
7
5
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A factor graph (bipartite).
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Relationship to GBP
Equivalent to GBP with a region graph that uses 5
copies of each plane. (Permitted because
redundant parity checks dont change problem.)



Other region graphs may give better performance,
but will not be as efficient.
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Speculations for the Future

• Problems on the AWGN channel may be caused by the
  fact that our algorithm creates GPC matrices
  where auxiliary bits talk almost exclusively to
  each other.
• With many redundant checks, its not clear one
  should give all messages equal weight, as in BP.
• Perhaps a better approach would be a
  message-passing algorithm that mimics multi-step
  majority logic decoding.

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Summary

• Even for a fixed code, finding a good GPC
  representation for BP decoding is an interesting
  problem.
• Classical codes are often decodable using BP, and
  have some advantages.
• We have presented a particular algorithm for
  transforming GPC matrices. It works well on the
  BEC, but there is still room for improvement on
  the AWGN channel.