Low-Density Parity Check Codes for Error Correction in Nanoscale ...
Low Density Parity Check Codes An Introduction
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Transcript of Low Density Parity Check Codes An Introduction
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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
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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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