Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes
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Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes K. Kasai and K. Sakaniwa Tokyo Institute of Technology
Aug. 2, 2011, ISIT @ st. petersburg
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1 / 20
LDPC convolutional codes
LDPC codes defined by sparse band parity-check matrices.
Memory saving window-decoding is possible.
Exhibit better asymptotic decoding performance than block codes.
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2 / 20
Regular LDPC codes for BEC
Hblk =
MAP BP blk ≈ 0.4294 < blk ≈ 0.48815 .
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3 / 20
¨ Felstrom-Zigangirov Construction
Hblk = 1. Divide the parity-check matrix like this.
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.
.
.
.
.
4 / 20
¨ Felstrom-Zigangirov Construction
Hconv = 2. Copy
many (2L + 1) times.
3. Put the copies at diagonal position. 4. We obtain a sparse band parity-check matrix.
MAP BP conv ≈ blk ≈ 0.48815 [Lentmaier et al.] .
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5 / 20
LDPC Convolutional Codes Excellent Performance
MAP Kudekar et al. proved BP conv = blk .
Kudekar et al. reported empirical evidence that this phenomenon also occurs for BMS channels other than BEC.
Many capacity-achieving results have been reported for memory channels, multi-access channels, rate-less coding, compressed sensing, relay channels, lattice coding, CDMA systems, K-SAT, quantum codes, Slepian-Wolf coding,
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6 / 20
Good points of spatially-coupled LDPC codes
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. . Minimum distance and water-fall trade-off .. Irregular LDPC codes often suffer from the dilemma between minimum distance and water-fall performance. Spatially-coupled LDPC codes have, at the same time, large minimum distance and .capacity-achieving water-fall performance. .. . . Universality .. Sason et al. proved that, in the limit of large column-weight and row-weight, regular LDPC codes achieve the capacity of any BMS channels under ML decoding. Hence, it is expected that .spatially-coupled LDPC codes universally achieve the capacity. .. .
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7 / 20
Drawback of spatially-coupled LDPC codes
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. . Drawback .. Infinite column-weight and row-weight are required to strictly achieve .the capacity, which leads to infinite computation per iteration. .. .
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. . Aim of this study .. Construct spatially-coupled LDPC codes which (universally) achieve the capacity under BP decoding with bounded column-weight and row-weight. . .. .
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.
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8 / 20
Gap to capacity and density of codes
Define “density” as the number of non-zero entries in the parity-check matrix of the code per information bits. E.g. (3,6) codes have density 6.
Sason and Urbanke showed the codes achieving the fraction 1 − of capacity under MAP decoding have density at least O(ln 1 )．
This result is valid only for the codes without punctured bits.
.
.
.
.
.
.
9 / 20
Capacity-Achieving Codes with Bounded Density Murayama et al. Tanaka and Saad observed by replica method that MN codes were capacity-achieving under ML decoding with bounded density for BSC and AWGNC. .
(l, r , g)-MacKay-Neal (MN) Codes
.
..
l r g g :punctured bit node
.
r l l(r + g) = r
RMN =
density: dMN
(r =g=2) .
=
.
4 RMN .
.
.
.
10 / 20
.
coding cate:
Capacity-Achieving Codes with Bounded Density Hsu and Anastasopoulos proved HA codes achieve the capacity of any BMS channels under ML decoding with bounded density. .
(l, r , g)-Hsu-Anastasopoulos (HA) Codes
.
..
r l g g
:punctured bit node l r r (1 + g + l) = r −l
coding rate: RHA = 1 −
(l=g=2) .
=
.
5 RHA .
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.
11 / 20
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.
density: dHA
Summary
. Two facts .. The BP threshold of LDPC convolutional codes is the MAP threshold of the constituent LDPC block codes.
.
. Do spatially coupled MN or HA codes achieve the capacity with bounded density under BP decoding? . ..
.
.
.
.
.
It is known that MN codes and HA codes achieve the capacity under MAP decoding with bounded density.
. .
.
.
. ..
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.
12 / 20
Results Couping MN codes
.. For any l, r , g ≥ 2, we observed the BP thresholds of spatially coupled (l, r , g)-MN codes are very close to the Shannon limit for the BEC. The minimum density 4/R is attained with r = g = 2. . .. .
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Protograph of spatially coupled (8,4,5)-MN code .
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13 / 20
Results (Cont.) Couping HA codes
.. For any l, r , g ≥ 2, we observed the BP thresholds of spatially coupled (l, r , g)-HA codes are very close to the Shannon limit for the BEC. The minimum density 5/R is attained with l = g = 2. . .. .
Protograph of spatially coupled (4,8,5)-HA code .
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14 / 20
EBP curve (wiggles are visible when w = 2)
The EBP EXIT curve (, hEBP ) and (xi , hEBP ) of the (l = 4, r = 2, g = 3, L = 16, w = 2)-MN code ensemble. .
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15 / 20
EBP curve (wiggle are invisible when w = 4)
The EBP EXIT curve (, hEBP ) and (xi , hEBP ) of the (l = 4, r = 2, g = 3, L = 16, w = 4)-MN code ensemble. .
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16 / 20
Duality
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EBP EBP hMN () = 1 − hHA (1 − )
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The generator matrix of an MN is identical to the parity-check matrix of a HA code. (l, r , g)-MN codes are dual codes of (r , l, g)-HA codes . . EBP duality ..
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17 / 20
Duality: EBP curve of uncoupled MN codes 1.0
1.0 l=4 r=2 g=2
0.8
0.8
0.7
0.7
0.6
0.6
0.5 0.4
0.4 0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
0.0 0.0
1.0
1.0
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
1.0
0.7
0.6
0.6
hEBP(ε)
0.8
0.7
0.5 0.4
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
l=6 r=3 g=3
0.9
0.8
0.0 0.0
0.1
1.0 l=6 r=3 g=2
0.9
hEBP(ε)
0.5
0.3
0.0 0.0
l=4 r=2 g=3
0.9
hEBP(ε)
hEBP(ε)
0.9
0.9
1.0
0.0 0.0
0.1 .
0.2
0.3 .
0.4 .
0.5 ε
0.6 .
0.7
0.8 .
0.9
1.0 .
18 / 20
Duality: EBP curve of uncoupled HA codes 1.0
1.0 l=2 r=4 g=2
0.8
0.8
0.7
0.7
0.6
0.6
0.5 0.4
0.4 0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
0.0 0.0
1.0
1.0
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
1.0
0.7
0.6
0.6
hEBP(ε)
0.8
0.7
0.5 0.4
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
l=3 r=6 g=3
0.9
0.8
0.0 0.0
0.1
1.0 l=3 r=6 g=2
0.9
hEBP(ε)
0.5
0.3
0.0 0.0
l=2 r=4 g=3
0.9
hEBP(ε)
hEBP(ε)
0.9
0.9
1.0
0.0 0.0
0.1 .
0.2
0.3 .
0.4 .
0.5 ε
0.6 .
0.7
0.8 .
0.9
1.0 .
19 / 20
Summary and Future works
Summary We observed that the BP threshold of spatially coupled MN and HA codes are very close to the Shannon limit with bounded degree. Ongoing works Simplify the proof of Kudekar et al. Construct dual capacity-achieving spatially-coupled code pair.
This slide will be available at Kasai’s web page. Google “Kenta Kasai.”
.
.
.
.
.
.
20 / 20
.
.
. ..
.
Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes K. Kasai and K. Sakaniwa Tokyo Institute of Technology
Aug. 2, 2011, ISIT @ st. petersburg
.
.
.
.
.
.
1 / 20
LDPC convolutional codes
LDPC codes defined by sparse band parity-check matrices.
Memory saving window-decoding is possible.
Exhibit better asymptotic decoding performance than block codes.
.
.
.
.
.
.
2 / 20
Regular LDPC codes for BEC
Hblk =
MAP BP blk ≈ 0.4294 < blk ≈ 0.48815 .
.
.
.
.
.
3 / 20
¨ Felstrom-Zigangirov Construction
Hblk = 1. Divide the parity-check matrix like this.
.
.
.
.
.
.
4 / 20
¨ Felstrom-Zigangirov Construction
Hconv = 2. Copy
many (2L + 1) times.
3. Put the copies at diagonal position. 4. We obtain a sparse band parity-check matrix.
MAP BP conv ≈ blk ≈ 0.48815 [Lentmaier et al.] .
.
.
.
.
.
5 / 20
LDPC Convolutional Codes Excellent Performance
MAP Kudekar et al. proved BP conv = blk .
Kudekar et al. reported empirical evidence that this phenomenon also occurs for BMS channels other than BEC.
Many capacity-achieving results have been reported for memory channels, multi-access channels, rate-less coding, compressed sensing, relay channels, lattice coding, CDMA systems, K-SAT, quantum codes, Slepian-Wolf coding,
.
.
.
.
.
.
6 / 20
Good points of spatially-coupled LDPC codes
.
. . Minimum distance and water-fall trade-off .. Irregular LDPC codes often suffer from the dilemma between minimum distance and water-fall performance. Spatially-coupled LDPC codes have, at the same time, large minimum distance and .capacity-achieving water-fall performance. .. . . Universality .. Sason et al. proved that, in the limit of large column-weight and row-weight, regular LDPC codes achieve the capacity of any BMS channels under ML decoding. Hence, it is expected that .spatially-coupled LDPC codes universally achieve the capacity. .. .
.
.
.
.
.
.
.
.
7 / 20
Drawback of spatially-coupled LDPC codes
.
. . Drawback .. Infinite column-weight and row-weight are required to strictly achieve .the capacity, which leads to infinite computation per iteration. .. .
.
. . Aim of this study .. Construct spatially-coupled LDPC codes which (universally) achieve the capacity under BP decoding with bounded column-weight and row-weight. . .. .
.
.
.
.
.
.
8 / 20
Gap to capacity and density of codes
Define “density” as the number of non-zero entries in the parity-check matrix of the code per information bits. E.g. (3,6) codes have density 6.
Sason and Urbanke showed the codes achieving the fraction 1 − of capacity under MAP decoding have density at least O(ln 1 )．
This result is valid only for the codes without punctured bits.
.
.
.
.
.
.
9 / 20
Capacity-Achieving Codes with Bounded Density Murayama et al. Tanaka and Saad observed by replica method that MN codes were capacity-achieving under ML decoding with bounded density for BSC and AWGNC. .
(l, r , g)-MacKay-Neal (MN) Codes
.
..
l r g g :punctured bit node
.
r l l(r + g) = r
RMN =
density: dMN
(r =g=2) .
=
.
4 RMN .
.
.
.
10 / 20
.
coding cate:
Capacity-Achieving Codes with Bounded Density Hsu and Anastasopoulos proved HA codes achieve the capacity of any BMS channels under ML decoding with bounded density. .
(l, r , g)-Hsu-Anastasopoulos (HA) Codes
.
..
r l g g
:punctured bit node l r r (1 + g + l) = r −l
coding rate: RHA = 1 −
(l=g=2) .
=
.
5 RHA .
.
.
.
11 / 20
.
.
density: dHA
Summary
. Two facts .. The BP threshold of LDPC convolutional codes is the MAP threshold of the constituent LDPC block codes.
.
. Do spatially coupled MN or HA codes achieve the capacity with bounded density under BP decoding? . ..
.
.
.
.
.
It is known that MN codes and HA codes achieve the capacity under MAP decoding with bounded density.
. .
.
.
. ..
.
.
12 / 20
Results Couping MN codes
.. For any l, r , g ≥ 2, we observed the BP thresholds of spatially coupled (l, r , g)-MN codes are very close to the Shannon limit for the BEC. The minimum density 4/R is attained with r = g = 2. . .. .
.
.
.
Protograph of spatially coupled (8,4,5)-MN code .
.
.
.
.
.
13 / 20
Results (Cont.) Couping HA codes
.. For any l, r , g ≥ 2, we observed the BP thresholds of spatially coupled (l, r , g)-HA codes are very close to the Shannon limit for the BEC. The minimum density 5/R is attained with l = g = 2. . .. .
Protograph of spatially coupled (4,8,5)-HA code .
.
.
.
.
.
.
.
.
14 / 20
EBP curve (wiggles are visible when w = 2)
The EBP EXIT curve (, hEBP ) and (xi , hEBP ) of the (l = 4, r = 2, g = 3, L = 16, w = 2)-MN code ensemble. .
.
.
.
.
.
15 / 20
EBP curve (wiggle are invisible when w = 4)
The EBP EXIT curve (, hEBP ) and (xi , hEBP ) of the (l = 4, r = 2, g = 3, L = 16, w = 4)-MN code ensemble. .
.
.
.
.
.
16 / 20
Duality
. ..
EBP EBP hMN () = 1 − hHA (1 − )
.
.
.
.
.
.
.
The generator matrix of an MN is identical to the parity-check matrix of a HA code. (l, r , g)-MN codes are dual codes of (r , l, g)-HA codes . . EBP duality ..
.
17 / 20
Duality: EBP curve of uncoupled MN codes 1.0
1.0 l=4 r=2 g=2
0.8
0.8
0.7
0.7
0.6
0.6
0.5 0.4
0.4 0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
0.0 0.0
1.0
1.0
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
1.0
0.7
0.6
0.6
hEBP(ε)
0.8
0.7
0.5 0.4
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
l=6 r=3 g=3
0.9
0.8
0.0 0.0
0.1
1.0 l=6 r=3 g=2
0.9
hEBP(ε)
0.5
0.3
0.0 0.0
l=4 r=2 g=3
0.9
hEBP(ε)
hEBP(ε)
0.9
0.9
1.0
0.0 0.0
0.1 .
0.2
0.3 .
0.4 .
0.5 ε
0.6 .
0.7
0.8 .
0.9
1.0 .
18 / 20
Duality: EBP curve of uncoupled HA codes 1.0
1.0 l=2 r=4 g=2
0.8
0.8
0.7
0.7
0.6
0.6
0.5 0.4
0.4 0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
0.0 0.0
1.0
1.0
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
0.9
1.0
0.7
0.6
0.6
hEBP(ε)
0.8
0.7
0.5 0.4
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1 0.1
0.2
0.3
0.4
0.5 ε
0.6
0.7
0.8
l=3 r=6 g=3
0.9
0.8
0.0 0.0
0.1
1.0 l=3 r=6 g=2
0.9
hEBP(ε)
0.5
0.3
0.0 0.0
l=2 r=4 g=3
0.9
hEBP(ε)
hEBP(ε)
0.9
0.9
1.0
0.0 0.0
0.1 .
0.2
0.3 .
0.4 .
0.5 ε
0.6 .
0.7
0.8 .
0.9
1.0 .
19 / 20
Summary and Future works
Summary We observed that the BP threshold of spatially coupled MN and HA codes are very close to the Shannon limit with bounded degree. Ongoing works Simplify the proof of Kudekar et al. Construct dual capacity-achieving spatially-coupled code pair.
This slide will be available at Kasai’s web page. Google “Kenta Kasai.”
.
.
.
.
.
.
20 / 20
