1. Low Density Parity Check codes
Performance similar to turbo codes
Do not require long interleaver to achieve good performance
Better block error performance
Error floor occurs at lower BER
Decoding is not trellis based
Iterative decoding
More iterations than turbo decoding
Simpler operations in each iteration step
Not covered by patents!

2. Introduction A binary parity check matrix Hn-kn
C = { vGF(2)n : vHT = 0 }
(Regular) LDPC code: The null space of a matrix H which has the following properties:
Its J rows are not necessarily linearly independent
 1s in each row
 1s in each column
No pair of columns has more than one common 1
 and  are small compared to the number of rows (J) and columns (n) in H
Density r = /n = /J
Irregular: The number of 1s in rows/columns may vary

3. Example
J = n = 15
 =  = 4

4. Gallager’s LDPC codes Select  and 
Form a kk matrix H consisting of  submatrices H1,...,H, each of dimensions kk
H1 is a matrix such that its ith row has its  1s confined to columns (i-1)+1 to i.
The other submatrices are column permutations of H1
The actual column permutations define the properties of the code

5. Gallager’s LDPC codes: Example
Select  = 4,  = 3
Column permutations for H2 and H3 selected so that no two columns have more than one 1 in common
(20,7,6) code

6. Rows and parity checks Each row of H forms a parity check
Let Al be the set of rows {h1(l),...,h(l)} that have a one in position l, i. e. the set of parity checks on symbol l.
Since no pair of columns has more than one common 1, any code bit other than bit l is checked by at most one of the rows in Al.
Thus there are  rows orthogonal on l, and any error pattern of weight  /2 can be decoded by one-step majority logic decoding. However, (especially when  is small) this gives poor performance
Gallager proposed two iterative decoding algorithms, one hard decision based and one soft decision based

7. Brief overview of graph theoretic notations
An undirected graph G(V,E)
Degree of a vertex/node : Number of edges adjacent to it
Adjacent edges: Connected to the same vertex
Path in a graph: Alternating sequence of vertices and edges (starting and ending in a vertex), where all vertices are distinct except for maybe the first and the last
A tree: A graph without cycles
Connected graph: There is a path between any pair of vertices
Bipartite graph: the set of vertices can be partitioned into two disjoint parts; so that each edge goes from a vertex in one part to a vertex in the other

8. Graphical description of LDPC codes
Tanner graph: A bipartite graph G(V,E), where
V = V1  V2
V1 is a set of n code-bit nodes v0,...,vn-1, each representing one of the n code bits
V2 is a set of J check nodes s0,...,sJ-1, each representing one of the J parity checks
There is an edge between v V1 and s V2 iff the code-bit corresponding to v is contained in the parity check s.
No cycles of length 4 for the Tanner graph of an LDPC code

9. Example

10. Decoding of LDPC codes
Several decoding techniques available. Listed in order of increasing complexity (and improving performance):
Majority logic decoding (MLG)
Bit flipping (BF)
Weighted bit flipping (WBF)
Iterative decoding based on belief propagation (IDBP) also known as the sum-product algorithm (SPA)
A posteriori probability decoding (APP)

11. Decoding notation
Codeword = channel input : v = (v0,..., vn-1), binary
Assume BPSK modulation on AWGN channel
Channel output : y = (y0,..., yn-1)
Hard decisions : z = (z0,..., zn-1), zj = 1 (0) yj > (<) 0
Rows of H : h1,..., hJ
Syndrome : s = (s1,..., sJ) = z  HT , sj = z  hj = l=0...n-1zlhj,l
Parity failure : at least one syndrome bit nonzero
Error pattern e = v + z
Syndrome : s = (s1,..., sJ) = e  HT , sj = e  hj = l=0...n-1elhj,l

12. Majority logic decoding
One-step MLG (see Chapter 8*)
Let Al be the set of rows {h1(l),...,h(l)} that have a one in position l, i. e. the set of parity checks on symbol l.
Set of  parity checks orthogonal on code bit l:
Sl = { s = z  h = e  h = l=0...n-1elhl : h Al }
If a majority of the parity checks on bit l are satisfied; conclude that el = 0. Otherwise, assume that el = 1.
This decoding is guaranteed to result in a correct decoding if at most /2 errors occur.
Problem:  is by assumption a small number.

13. Bit flipping decoding Introduced by Gallager in 1961
Hard decision decoding: First, form the hard decisions z = (z0,..., zn-1)
Compute the parity check sums s = (s1,..., sJ) = z  HT , sj = z  hj = l=0...n-1zlhj,l
For each bit l, find fl = the number of failed parity checks for bit l
Let S = the set of bits for which fl is large
Flip the value of the bits in S
Repeat from a) until all parity checks are satisfied, or a maximum number of iterations have been reached

14. Bit flipping decoding (continued)
Let S = the set of bits for which fl is large
For example:
S = the set of bits for which fl exceeds some threshold  that depends on the code parameters: , , minimum distance and channel SNR. If decoding fails, try again with reduced value of 
Simple alternative: S = the set of bits for which fl is maximum
Comments:
If few errors occur, decoding should commence in a few iterations
On a poor channel, the number of iterations should be (allowed to grow) large
Improvement: Adaptive thresholds

15. Weighted MLG and BF decoding
Use weighting to include soft decision/reliability information in the decoding decision
Hard decision decoding: First, form the hard decisions z = (z0,..., zn-1)
Weight: | yj|(l)min = min{| yi| : 0in-1, hj,i =1, h Al }
The reliability of the jth parity check on l
Weighted MLG: Hard decision on
El = s Sl (2s-1)| yj|(l)min
...where Sl = { s = z  h = e  h = l=0...n-1elhl : h Al }
Weighted BF:
Compute the parity checks s = (s1,..., sJ) = z  HT
For each bit l, compute El .
Flip the value of the bit for which El is largest
Repeat from a) until all parity checks are satisfied, or a maximum number of iterations have been reached

16. The sum-product decoding algorithm
Iterative decoding based on belief propagation
Symbol-by-symbol SISO: The goal is to compute
P( vl | y )
L(vl) = log ( P( vl = 1| y )/P( vl = 0| y ) )
A priori values: plx = P( vl = x), x{0,1}
qj,lx,(i) = P(vl = x |check sums  Al \ {hj} at ith iteration}
hj = (hj,0, hj,1, ..., hj,n-1); support of hj = B(hj) = { l : hj,l = 1}
j,lx,(i) = P(sj=0| vl = x,{vt:tB(hj)\{l}})   tB(hj)\{l} qj,tvt,(i)
qj,lx,(i+1) = j,l(i+1)  ht  Al \ {hj} t,lx,(i)
j,l(i+1) chosen so that qj,l0,(i+1) +qj,l1,(i+1) =1
P (i)(vl = x | y )=l(i)  hj  Al j,lx,(i-1)
 l(i) chosen so that P (i)(vl = 0| y ) + P (i)(vl = 1 | y )=1

17. The SPA on the Tanner graph
qj,lx,(i) = P(vl = x |check sums  Al \ {hj} at ith iteration}
j,lx,(i) = P(sj=0| vl = x,{vt:tB(hj)\{l}})   tB(hj)\{l} qj,tvt,(i)
qj,lx,(i+1) = j,l(i+1)  ht  Al \ {hj} t,lx,(i) + vl sj
j,lx,(i)
a,lx,(i)
b,lx,(i)
qj,mx,(i+1)
qj,lx,(i+1)
qj,nx,(i+1)
qj,lx,(i)
qj,mx,(i)
qj,nx,(i)

18. Suggested exercises