Download i+1

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
x,(i+1
qqj,lj,lx,(i)
j,lx,(i)
)
+
sj
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The sum-product decoding algorithm
• Initialization
• i=0
• For each pair (j,l) such that hj,l = 1,
• qj,l0,(i) = pl0 ,qj,l1,(i) = pl1
• (Assume implicit message passing)
a) For each l and for each hj Al compute j,lx,(i) ), x{0,1}
b) For each j and for each l such that hj Al compute qj,lx,(i+1) ,
and P (i+1)(vl = x | y ), for x{0,1}.
c) Determine the hard decisions based on P (i+1)(vl = x | y ) for
each l, and form the syndromes. If there remain parity
check failures and if iImax, set i=i+1 and repeat from a)
d) Output hard decisions based on the previous step.
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Complexity of the SPA
• Number of multiplications per iteration:
• O(2J+4n)
• Linear in the number of 1-entries in H
• Number of logarithm operations per iteration:
• O(n)
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Alternative SPA description
• Update extrinsic information at each decoding iteration
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Alternative SPA description
• Modified channel value matrix
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Design of LDPC codes
Requirements:
• Satisfy conditions of the LDPC definition
• Provide good performance when used with a specified decoder
• Random codes: Determine the connections of the bipartite Tanner
graph by using a (pseudo)random algorithm observing the degree
distribution of the code-bit vertices and the parity check vertices
• Regular
• Irregular
• Graph theoretic codes
• Combinatorial codes
• Codes from finite geometries
• Other algebraic constructions
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An introduction to Euclidean geometries
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The m-dimensional Euclidean geometry over GF(2s) EG(m, 2s)
consists of the set of all m-tuples (a0,...,am-1) such that each
component ai  GF(2s)
Each m-tuple (a0,...,am-1) is called a point
If a = (a0,...,am-1), then the set of 2s points {a: GF(2s)} is a line
which contains the origin (0,...,0)
Let L = {a: GF(2s)} be a line and b be a point not on L. We say
that a and b are linearly independent. In general, a line in EG(m, 2s)
is given by {b+a: GF(2s)} .
Two linearly independent points are connected by exactly one line
Two different lines have either zero or one point in common
Given a point a  EG(m, 2s) there are exactly (2ms -1)/(2s -1) lines
that intersect in the point a
The total number of lines in EG(m, 2s) is 2(m-1)s(2ms -1)/(2s -1) 8
Type-I geometry-Q LDPC codes
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Q: a finite geometry with n points p1,...,pn and J lines L1,...,LJ :
• Each line has  points
• Each point lies on  lines L1,...,LJ
• Two points are connected by exactly one line
• Two lines are either disjoint or they intersect at exactly one point
Incidence vector of L, vL=(v1,...,vn)  GF(2) n : vi = 1 iff pi L
Then w(vL) = 
HQ(1) = Jn matrix with
• Rows: the incidence vectors of all lines in Q : row weight 
• Columns corresponding to the n points of Q : column weight 
Then no pair of rows have more than one 1 in common
And no pair of columns have more than one 1 in common
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Type-I geometry-Q LDPC code.
Type-I geometry-Q LDPC: Properties
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Minimum distance   + 1
Tanner graph:
• Regular; Each of the n code bit nodes have degree  and each of
the J check nodes have degree  .
• No cycles of weight 4
• Does contain cycles of weight 6 (Exercise 17.23)
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Type-II geometry-Q LDPC codes
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Q: a finite geometry with n points p1,...,pn and J lines L1,...,LJ :
• Each line has  points
• Each point lies on  lines
• Two points are connected by exactly one line
• Two lines are either disjoint or they intersect at exactly one point
Intersecting vector of p, vp=(v1,...,vJ)  GF(2)J : vi = 1 iff p Li
Then w(vL) = 
HQ(2) = [HQ(1)]T= nJ matrix with
• Rows: intersecting vectors of all points in Q : row weight 
• Columns: incidence vectors of all lines in Q : column weight 
• Then no pair of rows have more than one 1 in common
• And no pair of columns have more than one 1 in common
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Type-II geometry-Q LDPC code. Companion of Type I code
Q = EG(m,
s
2)
: Type I
• n = 2ms points = number of code bits
• J = 2(m-1)s(2ms -1)/(2s -1) lines = number of parity checks
•  = 2s points on each line
• Each point is contained in  = (2ms-1)/(2s-1) lines
• Therefore the minimum distance is at least +1 = (2ms-1)/(2s-1)+1
Example: Q = EG(2, 24)
• n = 2ms points = 256
• J = 2(m-1)s(2ms -1)/(2s -1) lines = 272
•  = 2s = 16 points on each line
•  = (2ms-1)/(2s-1) = 17 minimum distance at least 18 (in fact =18)
• The rank of HQ(1) = 81, so the dimension is 175
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Q = EG(m,
s
2)
: Type II
• J = 2(m-1)s(2ms -1)/(2s -1) = number of code bits
• n = 2ms = number of parity checks
•  = (2ms-1)/(2s-1) Each point is contain
•  = 2s
• Therefore the minimum distance is at least  +1 = 2s +1
Example: Q = EG(2, 24)
• J = 2(m-1)s(2ms -1)/(2s -1) points = 272
• n = 2ms lines = 256
•  = (2ms-1)/(2s-1) = 17 points on each line
•  = 2s = 16 minimum distance at least 17 (in fact =17)
• The rank of HQ(1) = 81, so the dimension is 191
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Cyclic EG-LDPC codes
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Remove origin point and all lines passing through it
J0 = (2(m-1)s-1 )(2ms -1)/(2s -1) lines that do not pass through the origin
n0 = 2ms -1 points
 = 2s points on each line
0 = (2ms-1)/(2s-1)-1 lines through each point
Type I code: Minimum distance at least 0 +1= (2ms-1)/(2s-1)
This code, CEG,c(1)(m,0,s), turns out to be a cyclic code:
The (0,sth)-order EG code (Ch. 8*), generator polynomial gEG,c(1)(X)
 primitive element of GF(2ms)
h root of gEG,c(1)(X) for h < 2(m-1)s+ 2(m-2)s+1+1  minimum distance
Tighter than the 0 +1 bound except for m=2
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Cyclic EG-LDPC codes: Type I, m=2
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n = 22s – 1 points, J = 22s – 1 parity checks
Can show: n – k = 3s – 1 parity check bits
Dimension k = 22s – 3s
Minimum distance at least  +1 = 2s + 1
Density r = 2s / (22s – 1)
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Projective Geometry LDPC codes
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LDPC codes can also be designed using projective geometries in the
place of Q
Slightly different parameters
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Example: Type-I cyclic EG-LDPC
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m=2, s=5: (1023,781,33) code, R=0.763
Also shown: Type-I PG-LDPC (1057,813,34), R=0.769
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Example: Type-I cyclic EG-LDPC
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m=2, s=6: (4095,3367,65) code, R=0.83
Also shown: Type-I PG-LDPC (4161,3431,66), R=0.825
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Example: Type-I cyclic EG-LDPC
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m= s=3: Type I: a (511,139,79) code, R=0.272
Type-II EG-LDPC (4599,4227, 9), R=0.92
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Yet another example
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Type I cyclic EG-LDPC (255,175) code
Type I cyclic PG-LDPC (273,191); two random (273,191) codes
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FG LDPC codes vs. random
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Longer codes: Random LDPC codes should outperform FG LDPC
codes.
Encoding may be difficult in random codes due to lack of structure
Iterative encoding?
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Performance vs. number of iterations
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Suggested exercises
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17.12-17.14,17.23
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