Outline

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Local Theory of BER for LDPC Codes Instantons on
a Tree
Vladimir Chernyak Department of Chemistry Wayne
State University
In collaboration with Misha Chertkov
(LANL) Misha Stepanov (LANL) Bane Vasic (Arizona)
Special thanks Fred Cohen (Rochester)
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Outline

• Introduction and terminology Linear Block and
  LDPC codes, parity checks and Tanner graphs
• Effective spin models for decoding sMAP and BP
  approaches
• Local and global structures of LDPC codes The
  role of trees
• Instanton (optimal fluctuation) approach to BER
• Low SNR case High-symmetry local instantons,
  Shannon transition
• High SNR case Low-symmetry global instantons
• From high to low SNR Instantons with
  intermediate symmetries
• Towards non-tree instantons High SNR case,
  quasi-instantons and related painted structures
• Summary and future plans

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Linear block codes (parity check representation)
Tanner graph
Parity check matrix
- spin variables
- set of constraints
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Linear block codes and Tanner graphs
connections
variable (bit) nodes
checking nodes
words (spin representation)
code words
Equivalent codes (gauge invariance)
Gauge group
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Effective spin models and decoding approaches
Stat Mech interpretation was suggested by N.
Sourlas (Nature 89)
Set of magnetic fields (measurement outcome)
log-likelihoods
sMAP decoding (gauge invariant)
Approximate gauge non-invariant schemes
auxiliary variables defined on connections
Gallager 63 Pearl 88 MacKay 99
magnetization a-posteriori log-likelihoods
Iterative belief propagation (BP)
Belief propagation (BP) equation
All three schemes are equivalent in the loop-free
case
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Post FEC bit-error rate (BER) and instantons
Probability of a measurement outcome
PDF of magnetization
Gaussian symmetric noise case
Probability of a bit error
SNR
Instanton (optimal fluctuation) approach PDF is
dominated by the most probable noise
configuration (saddle point)
Lagrange factor
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Geometry of Tanner graphs
Local structure Each node has a tree
neighborhood
Universal covering tree (similar to Riemann
surfaces)
fundamental group
free group with g generators
genus
Gauss-Bonnet theorem (Euler characteristic)
The covering tree is universal and possesses high
symmetry
local curvature
Graphs with constant curvature
Wiberg 95 Weiss 00
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BP iterative algorithm and decoding tree
Decoding tree for BP with the fixed number of
iterations
On a tree the auxiliary field can be defined in
variable nodes
the only in-bound nearest-neighbor checking node
The field that represents the history of
iterations
BP magnetization is represented by the fixed
point of BP equation (that coincides with sMAP)
on the decoding tree
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Tree instantons
shortest loop length (girth)
Local theory (no repetitions of magnetic fields)
Express magnetic fields in terms of magnetization
Effective action for an instanton problem
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High-symmetry low SNR local instantons
Symmetric instanton effective action
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High symmetry Shannon transition
corresponds to the maximum of
Shannons transition is a local property of a code
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Low-symmetry high SNR global instantons
Painted structure
High SNR instantons are associated with painted
structures
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Intermediate instantons with partially-broken
symmetry
Symmetry is described by partially-painted
structures
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Instanton phases on a tree
m4, n5, l4. Curves of different colors
correspond to the instantons/phases of different
symmetries.
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Full numerical optimization (no symmetry assumed)
Area of a circle surrounding any variable node is
proportional to the value of the noise in the
node.
m2 n3 l3
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NO MORE TREES
For higher SNR instantons reflect the global
geometry of the Tanner graph (loop structure)
How do instantons look in the high SNR limit?
We apply the concept of the covering (decoding)
tree
Wiberg 95 Weiss 00
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High SNR instantons for LDPC codes approximate
BP equations
Relevant multiplication operation
High SNR limit approximate formula
Reduced variables
Min-sum
Infinite SNR limit multiplication formula for
reduced variables
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High SNR instantons painted structure
representation
Painted structure
Discrete (Ising) variables
Expressions for magnetization
Quasi-instantons
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High SNR instantons pseudo-code word
representation
Successful (matched) competition of
two pseudo-code words
Quasi-instanton relation
If B is a stopping set (graph)
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Summary

• We have analyzed instantons for BER on trees
• Depending on SNR BER is dominated by instantons
  of different symmetry
• Shannon transition for an LDPC code is determined
  by local structure of the code (curvature)
• For BP iterative decoding we have identified
  candidates that dominate BER
• Adiabatic expanding of instantons from high to
  lower SNR

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Truth
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