Local Theory of BER for LDPC Codes Instantons

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

Outline • Introduction and terminology: Linear Block and LDPC codes, parity checks and Tanner graphs • Effective spin models for decoding: s. MAP 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, quasiinstantons and related painted structures • Summary and future plans

Linear block codes (parity check representation) Tanner graph Parity check matrix mod 2 - “spin” variables - set of constraints

Linear block codes and Tanner graphs connections variable (bit) nodes checking nodes words (spin representation) code words Equivalent codes (gauge invariance) Gauge group

Effective spin models and decoding approaches Stat Mech interpretation was suggested by N. Sourlas (Nature ‘ 89) Set of magnetic fields (measurement outcome) = log-likelihoods s. MAP decoding (gauge invariant) “Approximate” gauge non-invariant schemes Gallager ’ 63; Pearl ’ 88; Mac. Kay ’ 99 auxiliary variables defined on connections magnetization =a-posteriori log-likelihoods Iterative belief propagation (BP) Belief propagation (BP) equation All three schemes are equivalent in the loop-free case

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

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 Gauss-Bonnet theorem (Euler characteristic) local curvature Graphs with constant curvature genus The covering tree is universal and possesses high symmetry Wiberg ’ 95 Weiss ‘ 00

BP iterative algorithm and decoding tree Decoding tree for BP with the fixed number of iterations The field that represents the history of iterations On a tree the auxiliary field can be defined in variable nodes the only in-bound nearest-neighbor checking node BP magnetization is represented by the fixed point of BP equation (that coincides with s. MAP) on the decoding tree

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

High-symmetry low SNR local instantons Symmetric phase: at any node on the tree depends primarily on the generation (counted from the center) Symmetric instanton effective action for j=0, …, l-2

High symmetry: Shannon transition corresponds to the maximum of Shannon’s transition is a local property of a code

Low-symmetry high SNR global instantons Painted structure (i) Contains the tree center (ii) Together with a variable node contains all nearest-neighbor checking nodes (iii) Together with a checking node contains exactly one outbound nearest-neighbor checking node (iv) Minimal subgraph with these properties High SNR instantons are associated with painted structures

Intermediate instantons with partially-broken symmetry Low SNR (high temp) “ 0” “ 1” “ 3” “ 2” “ 4” Symmetry is described by partially-painted structures High SNR (low temp)

Instanton phases on a tree m=4, n=5, l=4. Curves of different colors correspond to the instantons/phases of different symmetries.

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. m=2 n=3 l=3

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

High SNR instantons for LDPC codes: approximate BP equations Relevant “multiplication” operation High SNR limit approximate formula Reduced variables Infinite SNR limit “multiplication” formula for reduced variables Min-sum

High SNR instantons: painted structure representation Discrete (Ising) variables Expressions for magnetization Quasi-instantons Painted structure

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)

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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