This document proposes and evaluates a modular lifting method to improve the cycle properties of quasi-cyclic low-density parity-check (QC-LDPC) codes. The method involves lifting the parity check matrix of a base graph by shifting the circulant submatrices by powers of a modular value. This approach is shown to reduce the number of harmful trapping sets compared to existing lifting methods, resulting in up to 0.3 dB gain in decoding performance based on simulations of 5G standards codes. The modular lifting allows for more flexible shifts of the variable and check node rings compared to previous approaches while still maintaining the quasi-cyclic structure of the codes.

1. Generalization of Floor Lifting for QC-LDPC Codes:
Theoretical Properties and Applications
Vasiliy Usatyuk, Sergey Egorov
South-West State University,
Kursk, L@Lcrypto.com
Ilya Vorobyev, Nikita Polyanskii
Institute for Information Transmission Problems,
Moscow, vorobyev.i.v@yandex.ru
Kazan, Russia
September 14 - 17, 2018
German Svistunov
Omsk State Technical University, Omsk, Russia
G.V.Svistunov@gmail

2. 1
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100111
101001
010011
H
1v 2v 3v 4v 5v 6v
1c
2c
3c
Low density parity-check codes (LDPC-codes), – block
linear code of dimension k and code words 𝑥 size n, defined
by parity-check matrix H with size (n-k)·n, which contain low
density-parity check.
Parity-check matrix of size (n-k)·n contain about n checks.
1 2 5
1 4 6
1 2 3 6
0
0
0
x x x
x x x
x x x x
  
  
   
Every row of parity-check matrix H define equation:
Low Density Parity-Check Codes
Gallager R.G., “Low-density parity-check codes”, IRE Trans. Inform. Theory, vol. IT-8, pp. 21-28,
Jan. 1962.
0T
Hx  (2)GF

3. 2
Tanner Graph– equivalent bipartite graph for the parity check matrix H
1v 3v2v 4v 5v 6v
1c 3c2cChecks (row)
VN (columns)
Tanner Graph. Cycles and Girth
Tanner R.M., “A recursive approach to low complexity codes”, IEEE Trans. Inform. Theory, IT-27,
pp. 533-547, September 1981.
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100111
101001
010011
H
1v 2v 3v 4v 5v 6v
1c
2c
3c

4. 3
Tanner Graph– equivalent bipartite graph for the parity check matrix H
1v 3v2v 4v 5v 6v
1c 3c2c
Cycle - closed simple way in Tanner-graph.
Example of cycle 4:c2 → v1 → c3 → v6 → c2
Girth – shortest cycles in Tanner-graph.
Checks (row)
VN (columns)
Tanner Graph. Cycles and Girth
Tanner R.M., “A recursive approach to low complexity codes”, IEEE Trans. Inform. Theory, IT-27,
pp. 533-547, September 1981.

5. 4
Quasi-Cyclic LDPC(QC-LDPC codes)- LDPC-codes which parity-
check matrix defined by structured block submatrix –
Circulant Permutation matrix.
Applied QC-LDPC codes allow to simplify analysis of code and
graph properties, increase throughput (parallelism inside
circulant), decrease complexity of hardware implementation
based on barrel shifter.
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100010
010001
010110
101001
H
1v 2v 3v 4v 5v 6v
1c
2c
3c
0 1 1
0 1 0
0 1 1
0 1 0
QC
I I I
H
I I I
   
      
4c
Quasi-Cyclic LDPC codes
( ) 2 2:Circulant PermutationMatrix CPM of size  0 1 11 0 0 1 0 0
, ,
0 1 1 0 0 0
I I I     
       
     
R. M. Tanner, D. Sridhara, T. Fuja, ”A class of group structured LDPC codes”, Proc. ICSTA 2001,
Ambleside, England, 2001

6. 5
1
1
,
0
( ) 0 mod ( )k k
i
j j k
k
l Z


 
QC-LDPC codes contain cycle of size if (circulant permutation matrix) shifts satisfy equationi2
Cycles at QC-LDPC Codes
Fossorier M.P.C., “Quasi-cyclic low-density parity-check codes from circulant permutation
matrices”, IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1788–1793, 2004.
For example for cycle 4, 𝑥 𝑖𝑠 𝑎 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑛𝑡 𝑠ℎ𝑖𝑓𝑡 𝑖𝑛 𝑚𝑎𝑡𝑟𝑖𝑥:
0 0 1
0 0 1
, 2 2QC
I I I
H where I CPM of size
I I I
 
  
 
1 0 1 0 0 1
0 1 0 1 1 0
1 0 1 0 0 1
0 1 0 1 1 0
H
 
 
 
 
 
 
 (H )QCAuth size I
If you found cycles multiplied it to Authomorphism of Tanner graph, in our case Auth=2

7. 6
1
1
,
0
( ) 0 mod ( )k k
i
j j k
k
l Z


 
QC-LDPC codes contain cycle of size if (circulant permutation matrix) shifts satisfy equationi2
Cycles at QC-LDPC Codes
Fossorier M.P.C., “Quasi-cyclic low-density parity-check codes from circulant permutation
matrices”, IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1788–1793, 2004.
For example for cycle 4:

8. 7
Length adaptation of QC-LDPC Codes
We have protograph’s base matrix and circulant size .
Construct QC-LDPC code:
“Simple” target. with maximal girth
“Industrial” target. special structures of cycles
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1101000101011010
0110010010101011
0010101010100101
0001100101010101
0000110001101010
0000011010101010
0000001110010101
0000000101010101
protoH
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28380260003003902432070
061700360070310110202
00120220300120200030020
000232800250240509033
0000113700031220390310
000001820030029011080
000000331890011025014
00000002602808015033
H
9 circulant
2 -1 6 -1 7 -1 7 -1 8 -1 -1 -1 -1 -1 -1 -1
7 -1 6 -1 1 -1 -1 3 6 3 -1 -1 -1 -1 -1 -1
-1 2 -1 8 -1 2 -1 8 -1 1 8 -1 -1 -1 -1 -1
-1 5 -1 6 -1 2 4 -1 -1 -1 0 1 -1 -1 -1 -1
4 -1 6 -1 3 -1 1 -1 5 -1 -1 5 4 -1 -1 -1
7 -1 4 -1 -1 2 -1 0 -1 6 -1 5 -1 4 -1 -1
8 1 -1 4 -1 4 -1 5 -1 -1 8 -1 -1 2 7 -1
-1 1 -1 3 4 -1 5 -1 3 -1 -1 -1 4 -1 4 3
H
42 circulant size
8x16 protograph base matrix
Protograph
16 variable nodes
8 parity-check nodes
Code length N=16*42=672 Code length N=16*9=144

9. 8
On practice we shall store one parity-check
QC-LDPC code for maximal length
Pa33
Pa34 Pa3(n-
1) Pa3n
.
..
.
..
.
..
.
..
. . .
. . .
Py
0 0Pa3
I
... ... I... ...
Pa11
Pa12
Pa13
Pa14
Pa1(n-1)
Pa1n
Pa21
Pa22
Pa23
Pa24
Pa2(n-1)
Pa2n
.
.
.
.
.
.
Pam1
Pam2
. . .
. . .
Z3
Z1
L2
Z1
Lifting
One parity check matrix to support variable code length
Pa11
Pa12
Pa13
Pa14
Pa1(n-1)
Pa1n
Pa21
Pa22
...
...
Pam1
Pam2
. . .
Pam(n-
1) Pamn
Px
0 Pam
... 0
Pa33
Pa34
Pa3(n-1)
Pa3n
.
..
...
...
...
. . .
. . .
Py
0 0Pa3
I
... ... I... ...
Pa11
Pa12
Pa13
Pa14
Pa1n
Pa21
Pa22
Pa23
Pa24
Pa2n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Pam1
Pam2
Pam3
Pam4
Pamn
. . .
. . .
. . .
. . .
Z2
Z1
And from then get all shorter parity-check matrix

10. 9
On practice we shall store one parity-check
QC-LDPC code for maximal length
1 25 55 -1 47 4 -1 91 84 8 86 52 82 33 5 0 36 20 4 77 80 0 -1 -1
-1 6 -1 36 40 47 12 79 47 -1 41 21 12 71 14 72 0 44 49 0 0 0 0 -1
51 81 83 4 67 -1 21 -1 31 24 91 61 81 9 86 78 60 88 67 15 -1 -1 0 0
50 -1 50 15 -1 36 13 10 11 20 53 90 29 92 57 30 84 92 11 66 80 -1 -1 0
96H
And from then get all shorter lengths parity-check
matrix    
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 9624 HE
Z
Z
HE
upper
current
0 6 13 -1 11 1 -1 22 21 2 21 13 20 8 1 1 9 5 1 19 20 1 -1 -1
-1 1 -1 9 10 11 3 19 11 -1 10 5 3 17 3 18 1 11 12 1 1 1 1 -1
12 20 20 1 16 -1 5 -1 7 6 22 15 20 2 21 19 15 22 16 3 -1 -1 1 1
12 -1 12 3 -1 9 3 2 2 5 13 22 7 23 14 7 21 23 2 16 20 -1 -1 1
24H
Floor lifting
For example IEEE 802.16 QC-LDPC codes 4x24

11. 10
Proposed
Floor Scale Modular Lifting – Flow Chart
    
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 upperupper
upper
current
current zrHE
z
z
HE mod*
It further step to get flexible in broking VN
and CN ring-generalization of floor lifting.
Only ‘r’ (scale) optimization procedure is necessary for particular code

12. 11

13. 12
IEEE 802.16 (WiMAX), compare of Samsung’s(Floor Scale) and Floor Scale Modular Lifting

14. 13

15. 14
Code A is the QC-LDPC with base graph 2 (BG2) from 5G EMBB standard R1-1711982 Nokia
WF on LDPC parity check matrices 3GPP RAN1-NR2 Qingdao, China, 27th – 30th June 2017
Code B is BG2 lifted using proposed floor scale modular lifting

16. Summary
1. Proved probabilistic statement concerning a theoretical
improvement of proposed lifting method with respect to the
number of small cycles.
2. This approach improve cycle properties of QC-LDPC codes,
decrease number of harmful Trapping sets and show from 0.1
to 0.3 dB gain compare to EMBB lifting.
15

17. Thank You!

18. Modular lifting
:96)2(9624
:
22222
*
396
*
circulantwithmatrixcheckparityfromofpowertofrom
circulantflexibleusecanweHinvalueshiftonnrestrictiosomewithresultAs
ZZZZZZZ
asgroupcyclicrepresentcanWe


    currentuppercurrent zHEHE mod
If we collapse by power 2 CN’s and VN’s rings of QC-LDPC
It will be QC-LDPC too.
Seho Myung; Kyeongcheol Yang, "Extension of quasi-cyclic LDPC codes by lifting," in Information Theory, 2005. ISIT 2005. Proceedings.
International Symposium on, vol., no., pp.2305-2309, 4-9 Sept. 2005
Seho Myung; Kyeongcheol Yang; Youngkyun Kim, "Lifting methods for quasi-cyclic LDPC codes," in Communications Letters, IEEE , vol.10, no.6,
pp.489-491
gettowantyouwhichsizecirculantzcurrent 

19. 96,48,24:
22
*
2496
*
circulant
ZZZZ 
Floor lifting (heuristics approach nobody know why it work)
Floor incomprehensible operation under multiplicative group
   
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 upper
upper
current
current HE
z
z
HE
96,92,...,34,32,28,24:circulant
We broke rings of CN and VN at some place and it work
Seho Myung; Kyeongcheol Yang; Youngkyun Kim, "Lifting methods for quasi-cyclic LDPC codes," in Communications Letters, IEEE , vol.10,
no.6, pp.489-491
Aamod Khandekar, Thomas Richardson Qualcomm patent, US 8,578,249 B2
Floor lifting

20. Floor Lifting Example
1 25 55 -1 47 4 -1 91 84 8 86 52 82 33 5 0 36 20 4 77 80 0 -1 -1
-1 6 -1 36 40 47 12 79 47 -1 41 21 12 71 14 72 0 44 49 0 0 0 0 -1
51 81 83 4 67 -1 21 -1 31 24 91 61 81 9 86 78 60 88 67 15 -1 -1 0 0
50 -1 50 15 -1 36 13 10 11 20 53 90 29 92 57 30 84 92 11 66 80 -1 -1 0
   

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
 9624
96
24
HEHE
0 6 13 -1 11 1 -1 22 21 2 21 13 20 8 1 1 9 5 1 19 20 1 -1 -1
-1 1 -1 9 10 11 3 19 11 -1 10 5 3 17 3 18 1 11 12 1 1 1 1 -1
12 20 20 1 16 -1 5 -1 7 6 22 15 20 2 21 19 15 22 16 3 -1 -1 1 1
12 -1 12 3 -1 9 3 2 2 5 13 22 7 23 14 7 21 23 2 16 20 -1 -1 1
96H
24H

21. Repetitive whole lifting procedure is necessary to optimize a code

22. Lifted protograph minimum distance
upper bound estimation
Code family upper bound bigger or equal lifted
protograph upper bound
Performance of codes derived with
the same protograph
Lifting is critical for code
performance

23. Page 23
.
Ace Spectrum
60circulant
N=1440
Floor Scale Modular Lifting
Floor Modular Lifting

24. Code Instance Derived from Code Family
Protograph lifting algorithm

25. Page 25
.