Low Density Parity Check CodesAn Introduction

Yuta Toriyama

yuta@ee.ucla.eduAugust 24, 2012

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Channel Coding / FEC

Technique for controlling errors in transmission of data

Redundancy in error-correcting code

Hamming

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Simple Parity Check Example

Simple scheme: repeat each bit 3 times

Majority rule to recover single bit

(3,1) code, single error detection & correction

0 0 0 0

1 1 1 1

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Simple Parity Check Example

“Hypercube” representing gray code

0 0 0

1 1 1

1 1 0

1 0 0

0 0 1

0 1 0

0 1 1

1 0 1

0 0 0

1 1 1

1 1 0

1 0 0

0 0 1

0 1 0

0 1 1

1 0 1single bit error

correction

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Simple Parity Check Example

“Hypercube” representing gray code

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Linear Block Codes

Generator matrix G (n x m) Parity-check matrix H (m x k)

Generator matrix is a transformation from n to m dimensions

Codeword c is the nullspace (kernel) of the parity-check matrix

Example: Hamming(7,4)

c xG0 cH

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Low Density Parity Check Codes

Parity-check matrix is sparse

A particular LDPC code can be represented by a sparse bipartite graph (“Tanner graph”)

H is the bi-adjacency matrix of the Tanner graph

1

0

1

0

1

1

0

0

1

0

0

1

0

1

1

0

0

1

0

1

0

0

1

1

H =

1 2 3 4 5 6

1 2 3 4

variable nodes

check nodes

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Iterative Message Passing Algorithm

Iterative decoding algorithm v-nodes and c-nodes pass “messages” to each

other

Belief Propagation

1 2 3 4 5 6

1 2 3 4

variable nodes

check nodes

\

1 10 1 2 1

2 2j

ji i ji V i

r q

1 1 0ji jir r

\

0 1 1| 0i

ij ij i i j ij C j

q K pr c y r

\

1 1| 1i

ij ij i i j ij C j

q K pr c y r

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Non Binary LDPC

Numbers are limited elements in GF(2p)

Decoding is much more complicated

Performs better than binary LDPC, especially in the case of short or medium codeword lengths

Davey, M.C.; MacKay, D.; , "Low-density parity check codes over GF(q)," Communications Letters, IEEE , vol.2, no.6, pp.165-167, June 1998

Summary

LDPC codes show strong potential as a set of codes with very low BER

Computational complexity of decoding algorithms needs to be resolved to make NB-LDPC practical as well as allow for further constructive research

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