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)

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

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

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

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

Post FEC bit-error rate (BER) and instantons PDF of magnetization Probability of a measurement outcome Probability of a bit error Gaussian symmetric noise case Instanton (optimal fluctuation) approach: PDF is dominated by the most probable noise configuration (saddle point) SNR 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) genus local curvature Graphs with constant curvature 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 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

Tree instantons Local theory (no repetitions of magnetic fields) shortest loop length (girth) 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) for j=0,…,l-2 Symmetric instanton effective action

High symmetry: Shannon transition corresponds to the maximum of Shannon’s transition 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 “0” “2” “1” “4”“3” Low SNR (high temp) High SNR (low temp) Symmetry is described by partially-painted structures

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 We apply the concept of the covering (decoding) tree Wiberg ’95 Weiss ‘00 For higher SNR instantons reflect the global geometry of the Tanner graph (loop structure) How do instantons look in the high SNR limit?

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 Painted structure Discrete (Ising) variables Expressions for magnetization Quasi-instantons

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

Truth … menu main slide