Algebra LDPC Codes and

Algebra LDPC Codes and Semi-Algebra LDPC Codes

A B

CD

O180

Π-Rotation LDPC Code

Subash Shree Pokhrel

Email: [email protected]

2

Semi-Random LDPC Codes

Construction Approach

① Design a parity check matrix, H matrix

② Design a codeword c

pH

dp HHH

dH

d

p

c

cc

m×m square matrix (m is the number of rows)

m×n-m square matrix (n is the number of columns)

Original information bits

Parity check bits

3



③ LDPC Encoder

0

d

pdp

c

cHHcH

Codeword c does not have errors

ddppddppddpp cHHcvcHcHcHcH 1)(0

Mod 2 operation is considered

ddpdddp cHHccHHcodeword 2mod1))(()( 11

G matrix

4



DIH p

I : identity matrix; D: dual matrix

0000000000

1000000000

0100000000

0010000000

0001000000

0000100000

0000010000

0000001000

0000000100

0000000010

D

1000000000

0100000000

0010000000

0001000000

0000100000

0000010000

0000001000

0000000100

0000000010

0000000001

I

1000000000

1100000000

0110000000

0011000000

0001100000

0000110000

0000011000

0000001100

0000000110

0000000011

pH+ =

5


Construction Approach Right Structure Design

- based on random codes (Mackay) ⑴ regular column weight (j, integer) regular row weight (k, integer) ⑵

dj

d

d

H

H

H

H

2

1

110

11

01

6


Advantages

① Encoding is simpler than a full Gaussian Elimination (full-random LDPC codes);

② They can be design as any code rate codes;

③ It requires a little memories to store right

structure if right part is sparse when j is a small integer.

7

How to design a good right structure?

dH

Tradeoff between Simple Implementation and Best Performance

Question?

Goal

8

The Π-rotation are described by three set of integers for the classic version.

The three integers [m,a,b] are used to define permutation matrix. The permutation matrix has a single ‘one’ in each column and row of the matrix.

Its size is determined by the value of m.

Π-Rotation LDPC code

9

Π-Rotation LDPC Design


① Basic Element Design πA Algorithm

l=mfor j1=1 to m i=a*j1+b mod (m+1-j1) πA (j1)=m-si

for j2=i to l-1 sj2=sj2+1

next j2 l=l-1 next j1

Example

1. Key [m, a, b]=[6, 4, 3]2. S={s0, s1,……,sm-1} where s0=1,….sm-1=m-13. πA

000100

010000

001000

000010

000001

100000

A

10


Construction Approach ② Related Element Design πB,πC,πD

000100

010000

001000

000010

000001

100000

A

001000

000010

000100

010000

100000

000001

B

000001

100000

010000

000100

000010

001000

C

100000

000001

000010

001000

010000

000100

D

11



③ Whole Parity Check Matrix Design H

CBAD

BADC

ADCB

DCBA

H

1000

1100

00

0011

Example (every column from bottom)

541326

000100

010000

001000

000010

000001

100000

A 623145

001000

000010

000100

010000

100000

000001

B

12



④ Whole Parity Check Matrix Design H

CBAD

BADC

ADCB

DCBA

H

1000

1100

00

0011

]154632[]623145[]541326[]236451[

]623145[]541326[]236451[]154632[

]541326[]236451[]154632[]623145[

]236451[]154632[]623145[]541326[

1000

1100

00

0011

H

13


Minimum Distance mind

2

1mindCorrect 1min dDetect

Correct=1 Detect=3

2

1jCorrect 1 jDetect

Larger minimum distance get better performance

14

Semi-Algebra LDPC Codes


Basic Algebra Formulas

}1,,2,1,0{ tSi

)(mod1 tq k

},,,,{ 12 smsqsqsqs

)(mod tssq sm

Example (t=31, k=5, j=3)

s=1; ms=5; {1, 2, 4, 8, 16}s=3; ms=5; {3, 6, 12, 24, 17}s=5; ms=5; {5, 10, 20, 9, 18}

15


Whole H Matrix Design

11,1B

22,1B

43,1B

84,1B

165,1B

31,2B

62,2B

123,2B

244,2B

175,2B

51,3B

102,3B

203,3B

94,3B

185,3B

pH 9393dH 15593

H

)()()(

)()()(

)()()(

1000

100

00

0011

1,11,10,1

1,11,10,1

1,01,00,0

pBpBpB

pBpBpB

pBpBpB

H

kjjj

k

k

625.0155/931

1

/1

1

nmR

16


Minimum Distance mind

2

1mindCorrect 1min dDetect

Correct=3 Detect=7

2

12 jCorrect 12 jDetect

Larger minimum distance get better performance

17


Simulation Results

1.5 2 2.5 3 3.5 410

-4

10-3

10-2

10-1

Eb/No

BE

R

RC N=348

S.R N=348

18

References

[1] Rich Echard and S.C. Chang, The extended irregular –rotation LDPC codes, IEEE Communication Letters, Vol. 7, No. 5, May, 2003. Page(s): 230-232.

[2]Yu Yi and Moon Ho Lee, “Semi-Algebraic Low-Density Parity- Check Codes,” 3rd International Symposium on Turbo Codes, France, Sept 5th, 2003. Page(s): 379-382.

[3]Marc. Fossorier, “Quasi-Cyclic Low-Density Parity-Check Codes,” International Symposium on Information Theory, Japan, June 4th, 2003.

[4] Yu Yi and Moon Ho Lee, “Optimized Low-Density Parity Check (LDPC) Codes for Bandwidth Efficient Modulation”, IEEE VTC Fall 2003, Florida, USA.

[5] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399- 431, Mar. 1999.

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